Cech Closure Spaces: A Unified Framework for Discrete and Continuous Homotopy

Motivated by constructions in topological data analysis and algebraic combinatorics, we study homotopy theory on the category of Cech closure spaces $\mathbf{Cl}$, the category whose objects are sets endowed with a Cech closure operator and whose morphisms are the continuous maps between them. We introduce new classes of Cech closure structures on metric spaces, graphs, and simplicial complexes, and we show how each of these cases gives rise to an interesting homotopy theory. In particular, we show that there exists a natural family of Cech closure structures on metric spaces which produces a non-trivial homotopy theory for finite metric spaces, i.e. point clouds, the spaces of interest in topological data analysis. We then give a Cech closure structure to graphs and simplicial complexes which may be used to construct a new combinatorial (as opposed to topological) homotopy theory for each skeleton of those spaces. We further show that there is a Seifert-van Kampen theorem for closure spaces, a well-defined notion of persistent homotopy, and an associated interleaving distance. As an illustration of the difference with the topological setting, we calculate the fundamental group for the circle, `circular graphs', and the wedge of circles endowed with different closure structures. Finally, we produce a continuous map from the topological circle to `circular graphs' which, given the appropriate closure structures, induces an isomorphism on the fundamental groups.


Introduction
Homotopy theory has long been one of the primary tools used to study topological spaces, and generalizations of the theory have had dramatic implications in other areas as well, in particular in algebra and algebraic geometry. There have recently been a number of attempts to extend the reach of homotopy theory to more discrete geometrical objects, such as graphs [1], directed graphs [20], and simplicial complexes [2][3][4], and then to try to characterize combinatorial properties of these objects in terms of their discrete homotopy invariants. In parallel, a different approach to discretization is developed in [23], in which the homotopies themselves are discretized, and this is then used to show the existence of certain relations in the fundamental group of geodesic spaces. Ideas from algebraic topology are also being used to study spaces where the natural topologies available don't capture the desired topological picture. This occurs, for instance, when trying to infer information about the topology of a manifold given a set of points sampled from it, a problem which has motivated the development of topological data analysis [10,19,27]. (We remark in passing that, although it's true that there are many topologies on finite sets which have interesting homotopy groups, the neighborhoods in such topologies are typically unrelated to a metric on the set, which is undesirable when studying a set of points sampled from a metric space. For an extensive discussion of finite topological spaces, see [5].) In this article, we develop homotopy theory in the category of closure spaces, which, as we will show, allows for the application of homotopy theory in all of the situations mentioned above, and which additionally reduces to the standard theory for topological spaces. Although our approach is not necessarily equivalent to the ones cited earlier, it nonetheless produces a unified construction of homotopy theory in all of these different contexts, in addition to defining a non-trivial homotopy theory for point clouds. Historically, homotopies on closure spaces were first defined in [16], with the aim of developing an alternate version of shape theory, which was pursued in [17,18]. These homotopy groups were then applied to directed graphs in [15]. A construction of homotopy groups on limit spaces (and therefore also closure spaces) is also given in [24], Section A.2, but was not developed further. To the best of our knowledge, this article is the first to apply the homotopy of closure structures to finite metric spaces and general skeleta of simplicial complexes, as well as the first to advance the general theory further since the works cited above. The plan of the paper is as follows. In Section 2, we give a formal introduction to closure spaces, and we collect the statements of results about them that we will need in what follows, Section 3 gives details of a specific family of closure operators of interest on metric spaces, in particular for topological data analysis on point clouds, and in Section 4 we study covering spaces, the fundamental group of the circle endowed with different closure structures, closure operators on graphs and skeleta of simplicial complexes, and give a definition of 'persistent' homotopy and the related interleaving distance. In the remainder of this section, we introduce the core idea behind our approach in the context of point clouds.
Given a set of points sampled from a topological probability space X, it's natural to ask whether the topological invariants of X can be recovered from the sample. Most current attempts at doing so, in particular persistent homology and homological manifold learning, first assume that X has a metric structure, and then proceed by replacing the points with balls of varying radii, effectively thickening the original set of observations. In manifold learning [12,21,22], the aim is to prove that, under favorable conditions, i.e. with high probability given an appropriate radius and a dense enough sample, the union of the balls centered at the sample points is homotopy equivalent to the space from which the samples were drawn, or else that some topological invariants of the original space may be recovered. In persistent (Čech) homology [10,13,19], one attempts to recover topological information from a metric invariant built using a one-parameter family of unions of balls around the sample points.
Instead of approximating the target space with auxiliary topological spaces built on our finite sample, however, our point of departure is to ask what sort of homotopy theory one may construct directly on finite sets of points, and then to construct (weak) homotopy equivalences between the sets of samples and the spaces from which they're sampled. While this might at first appear to lead to a trivial theory -and indeed it does, if one stays within the category Top -what we find is that, by changing the category, we are able to develop a homotopy theory which provides information about the global configuration of sample points without invoking an intermediate topological space. This is accomplished by 'coarsening' the continuous maps rather than thickening the space, and, as we will see, this focus on the maps makes homotopy into a functor from the category of closure spaces to the category of groups. This, in particular, enables us to easily define a homomorphism between homotopy groups starting from a morphism in our category, something which remains difficult with other approaches, in particular for maps between metric spaces which do not preserve the metric.
We give a simple example to illustrate what is meant above by 'coarsening' a continuous map. Consider the spaces S 1 = [0, 1]/(0 ∼ 1) and X = 0, 1 5 , . . . , 4 5 ⊂ S 1 . We would like to know what properties of S 1 we can recover from properties of X. From a homotopical point of view, we would like to find a category in which there is an equivalence between S 1 and X that we may invert in the homotopy category, which means, at the very least, that we need to have non-trivial morphisms from S 1 to X, even though, topologically, the only continuous maps are the constant maps. In closure spaces, however, we have a new class of maps to consider. In Section 3, we show that, on a family of closure structures on metric spaces, there is an ǫ-δ version of continuity for these closure spaces, which we describe here. For a pair of non-negative real numbers (q, r), we say that a map f : S 1 → X is (q, r)continuous at a point x iff ∀ǫ > 0, ∃ δ(x) > 0 such that x − y < q + δ implies f (x) − f (y) < r + ǫ. According to this defintion, the 'nearest neighbor map' f : S 1 → X given by (1.1) f , for x ∈ 9 10 , 1 ∪ 0, 1 10 1 5 , for x ∈ 1 10 , 3 10 2 5 , for x ∈ 3 10 , 5 10 3 5 , for x ∈ 5 10 , 7 10 4 5 , for x ∈ 7 10 , 9 10 is (q, r)-continuous for r ≥ 1 5 and 0 ≤ q ≤ r − 1 5 , although it is clearly discontinuous topologically. To see this continuity, note that, seen as a subset of S 1 , each point in X is at most a distance of 1 10 from any given point in S 1 . We therefore have for any (q, r) with q + 1 5 ≤ r, where we let δ = ǫ. As we will see in Sections 3 and 4, we may also construct homotopies and homotopy groups so that, for a certain range of (q, r), both spaces have the same fundamental group as the (topological) circle. This example illustrates two important points. The first is that topologically continuous maps are, in some sense, too rigid to be useful for a homotopy theory of point clouds, and the second is that they are not the only choice.
We make three additional observations about the modified notion of continuity above. First, note that for the pair q = 0, r = 1 10 , the map f is not 0, 1 10continuous. In particular, this illustrates how modifying continuity in this way allows (topological) discontinuities at any point, but only if the jumps at (topologically) discontinuous points are controlled, the allowable size of the (topological) discontinuity being given by the pair (q, r).
Second, we observe that topological continuity does not imply (q, r)-continuity for all q and r. Consider, for instance, multiplication by 2 on the real line, i.e. f : R → R, f (x) := 2x. While clearly continuous topologically, f is not (q, r)continuous for any pair (q, r) where q > 1 2 r > 0. To see this, first observe that any interval I x,r,ǫ := (x − r − ǫ, x + r + ǫ) must necessarily be the image of the interval 1 2 (x − r − ǫ), 1 2 (x + r + ǫ) . It's therefore clear that, for q > 1 2 r and sufficiently small ǫ > 0, there is no δ > 0 for which the interval f x 2 − q − δ, x 2 + q + δ ⊆ I x,r,ǫ , and so f is not (q, r)-contiuous at the point x 2 . This illustrates that, while local discontinuities are allowed, a (q, r)-continuous function must be rather uniformly controlled at the scale which determines (q, r)-continuity. While at first perhaps unsettling, we will see that, for point clouds, this rigidity has the desirable effect of keeping nearby points near each other after the application of a (q, r)-continuous function.
Finally, we remark that (q, r)-continuous maps between metric spaces need not be coarse maps in the sense of coarse geometry. That is, at large enough scales, a (q, r)-continuous map may send points which are initially a finite distance apart arbitrarily far away from each other, so long as the local (q, r)-continuity condition is satisfied. As an example, take the set of points Define the function F : X → R 2 by F (a, b) := (a · b, 0). This function is (1, 1)continuous, but it is not a coarse map, since there is no uniform bound on the distance d(F (a, b), F (c, d)). Together, these examples illustrate the fundamental features of the maps between the closure structures on metric spaces that will be our main objects of interest: they are wild at small scales, rigid at medium scales, and flexible at large scales.

The category Cl
A closure structure on a space X collects all of the information about the neighborhoods of each point in X. It is weaker than a topology, but, as we will see, still allows for the construction of a rich homotopy theory which extends classical homotopy on the subcategory Top. Among other things, this will allow us to construct weak homotopy equivalences between spaces which are topologically very different, but where there exist closure structures with similar characteristics, giving rise to isomorphic homotopy groups. In this section, we define closure spaces, and we include the statements of a number of results which we will use in this paper, both as an introduction to the subject -which, while classical, is not broadly known -as well as for reference.
In the examples and computations in this article, we will be mainly concerned with a natural family of closure structures induced by a metric, but the theory holds unchanged in the general setting. For additional general results on closure spaces and for the proofs of the results given here, we refer the reader to the results on closure spaces in the book [11].
2.1. Čech closure spaces. Definition 2.1. Let X be a set. A Čech closure operator on X is a map c : A pair (X, c) is called a Čech closure space, or simply a closure space, and for a set A ⊂ X, we call c(A) the closure of A. If A = {x}, we will write c({x}) as c(x). We sometimes say that the map c gives a Čech closure structure on X.
Given two Čech closure operators c 1 and c 2 on the same space X, we say that c 1 is finer than c 2 , and c 2 is coarser than c 1 , iff, for each A ⊂ X, c 1 (A) ⊂ c 2 (A), and we write c 1 < c 2 to denote this relation.
Remark 2.2. Note that the definition above immediately implies that the closure operator is monotone, i.e. A ⊂ B =⇒ c(A) ⊂ c(B).
We will make use of the following special closure structures in the sections that follow. Definition 2.3. Let X be a set. If a closure structure c on X satisfies c(x) = x for every point x ∈ X, then we say that c is the discrete closure structure on X, or simply discrete. If a closure structure x on X satisfies c(A) = X for every nonempty A ⊂ X, then we say the c is the indiscrete closure structure on X, or simply indiscrete.
We may also construct closure spaces via interior operators, defined as follows.
Definition 2.4. Let X be a set. An interior operator on X is a map Int : P(X) → P(X) which satisfies Definition 2.5. Given a closure space (X, c) we define the operator i c : P(X) → P(X) to be i c (A) = X − c(X − A). Similarly, given a set X with an interior operator i, we define an operator c i : Proposition 2.6. For any closure space (X, c), i c is an interior operator on X, and c (ic) = c. Similarly, for any interior operator i on a set X, c i is a Čech closure operator on X, and i (ci) = i.
Proof. Given a closure operator c on X, it follows immediately from the axioms for closure operators and de Morgan's laws that i c is an interior operator. In the same way, given an interior i operator on X, c i is a closure operator by the axioms for interior operators and de Morgan's laws. Finally, for any A ⊆ X, we have Definition 2.7. Let (X, c) be a closure space. We say that a subset A ⊂ X is While open and closed sets exist in closure spaces, in the following they take a secondary role to the neighborhoods of a set, defined below, which are not necessarily open or closed.
Neighborhoods and open sets are related by the following theorem.
iff it is a neighborhood of all of its points, or, equivalently, if it is a neighborhood of itself.
Definition 2.10. Let (X, c) be a closure space. A base of the neighborhood system of A ⊂ X is a collection B of subsets of X such that each set B ∈ B is a neighborhood of A, and each neighborhood of A contains a set in B. A subbase of the neighborhood system of Y ⊂ X is a collection C of subsets of X such that the collection of all finite intersections of elements of C is a base of the neighborhood system of Y . When A contains only a single point x ∈ X, we will sometimes use the term local base (local subbase) at x to refer to the base (subbase) of the neighborhood system of {x}.
The next theorems show that we may also construct a closure structure on a set X by specifying the local bases at each point. We first note, however, that any local base satisfies the following Proposition 2.11 ([11], 14.B.5). Let (X, c) be a closure space. If U(x) is a local base at a point x, then the following conditions are satisfied: We now see how to construct closure structures on a set X by designating a special collection of sets for each point x ∈ X.
Theorem 2.12 ([11], Theorem 14.B.10). For each point x ∈ X, let U(x) be a collection of sets that satisfies conditions 1-3 in Proposition 2.11. Then there exists a unique closure structure c on X such that, for each x ∈ X, U(x) is a local base at x in (X, c). (1) For each element x ∈ X, let U(x) be a filter on X such that x ∈ U(x). Then there exists a unique closure structure c on X such that U(x) is the neighborhood system at x in (X, c) for every x ∈ X.
(2) For each element x ∈ X, let V(x) be a non-empty family of subsets of X such that x ∈ V(x). Then there exists a unique closure structure c on X such that V(x) is a local subbase at x in (X, c) for every x ∈ X.
The next theorem and corollary show the connection between the closure of a set and neighborhoods of points, generalizing familiar facts from topological spaces.
Theorem 2.14 ( [11], Theorem 14.B.6). Let (X, c) be a closure space. Given a set A ∈ X, a point x ∈ X is contained in c(A) iff every neighborhood U of x intersects A non-trivially. . Let (X, c) be a closure space. If U is a local base at a point x ∈ X, then for a subset A ⊂ X, x ∈ c(A) iff, for each U ∈ U, Remark 2.16. Note that, by Theorem 2.14 and Corollary 2.15, the closure operator c constructed in Theorem 2.12 and Corollary 2.13 must be of the form Example 2.18. Every topological space X is a closure space with the closure operator defined by c(A) =Ā. Note that for the closure operator on topological spaces, c(c(A)) = c(A). Furthermore, if (X, c) is a closure space with c 2 = c, then the collection of sets U := {U ⊂ X | X − U is closed, i.e. c(X − U ) = X − U } forms a topology. To see this, first note that ∅ and X are in U, since c(∅) = ∅, and X ⊂ c(X), c(X) ⊂ X =⇒ c(X) = X. Second, for an arbitrary collection of sets in U, say It follows that ∪ λ∈Λ U λ ∈ U. Finally, for a finite intersection of sets forms the open sets of a topology whose closed sets are the fixed points of the operator c. A closure operator satisfying c 2 = c is called a topological or Kuratowski closure operator.
Kuratowski closure operators are characterized by the following (4) For each x ∈ X, if U is a neighborhood of x, then there exists a neighborhood V of x such that U is a neighborhood of each point of V , i.e. every neighborhood of any point x ∈ X is a neighborhood of a neighborhood of x.
Remark 2.20. A pair (X, c X ) is sometimes called a pretopology in the literature. However, since these pairs, and not topologies, are our main objects of study, we have elected to revert to the older nomenclature used in [11], which we believe does not semantically relegate these spaces to secondary, or preparatory, status. This convention also has the advantage of making Čech closure structures terminologically distinct from pretopologies in the sense of Grothendieck, which are different objects altogether. In this article, a closure space will always refer to a space with a closure operator that satisfies the axioms in Definition 2.1, although the reader should be warned that there is some variation in the literature. In particular, except when explicitly stated, we will not require that c 2 = c, which, combined with the axioms in Definition 2.1, would make the closure operator into what is known as a Kuratowski closure operator. As Kuratowski closure operators induce a topology whose closed sets are the closed sets given by the operator, we will refer to spaces with Kuratowski closure operators simply as topological spaces.  We say that C is a cover of (X, c X ) if ∪ U∈C U = X, i.e. if every point x ∈ X is contained in some U ∈ C. We say that C is an interior cover of (X, c X ) if ∪ U∈C i c (U ) = X, i.e. if every point x ∈ X has a neighborhood in C.
(1) Any cover of a closure space (X, c) by open sets is an interior cover, which we call an open cover of (X, c).
(2) Let (V, E) be the graph defined by where the integers are to be understood modulo 4. Then the family (3) For the graph (V, E) above, the family of sets C = {{i−1, i, i+1} | i ∈ {0, 2}} is a cover of the set V , but C is not an interior cover of the closure space (V, c E ), since there are no neighborhoods of the vertices v = 1 or v = 3 in C.
Compactness in closure spaces now takes the following form.
Definition 2.23. We say that a closure space (X, c) is compact iff every interior cover of X has a finite subcover.
Remark. We note that the finite subcover in the Definition 2.23 above may not itself be an interior cover. We also remark that, while Definition 41.A.3 in [11] defines compactness on closure spaces in terms of filters, the above definition is equivalent by Theorem 41.A.9 in [11], and this one is the more useful version for the purposes of this article.
2.3. Continuous functions. Continuity for maps between closure spaces is defined as follows.
Definition 2.24. Let (X, c X ) and (Y, c Y ) be closure spaces. We say that a map f : We say that a function f is continuous iff f is continuous at every point x of X.
Equivalently, a function f is continuous iff for every set U ⊆ X, f (c(U )) ⊆ c(f (U )).
We present in this subsection several important basic results on continuous maps between closure spaces. The proofs and additional results may be found in [11], Section 16.A. We begin with a result showing that the composition of continuous functions is continuous.
In particular, the composition of two continuous maps is continuous.
The next proposition indicates when the identity is continuous, as a map on the same space with two different closure structures.
there exists a neighborhood U ⊂ X of x such that f (U ) ⊂ V . The function f is continuous iff the above is satisfied for every point x ∈ X.
The above result further specializes to open and closed sets.
In particular, this implies Corollary 2.29. Let (X, c) be a closure space whose only open sets are ∅ and X, and suppose that (Y, τ ) is a T 0 topological space. Then every continuous map f : (X, c) → (Y, τ ) is constant.
Proof. Suppose f is not constant. Then there are two points x, y ∈ X such that f (x) = f (y). Since Y is T 0 , every pair of points x, y ∈ Y , at least one of them has an open neighborhood U which does not contain the other point. Without loss of generality, let V ⊂ Y be a neighborhood of f (x) with f (y) / ∈ V . Then f −1 (V ) = ∅ or X, which is a contradiction, since, on the one hand, x ∈ f −1 (V ), so f −1 (V ) = ∅, and, on the other hand, y / ∈ f −1 (V ), so f −1 (V ) = X. Therefore, f is constant.
The following theorem and its corollaries allow us to paste together a continuous map f : (X, c) → (Y, c ′ ) from continuous maps on sets in covers of X.   . Let {U a } be an interior cover of a closure space (X, c), and let (Y, c ′ ) be a closure space. If f : X → Y is a map such that the restriction of f to each subspace (U a , c Ua ) is continuous, then f is continuous.

Topological modifications.
It will sometimes be useful to appeal to a topological closure structure 'generated by' a given closure structure on a closure space (X, c). We construct this as follows. Then τ c X is the finest topological closure coarser than c X , i.e. (τ c X ) 2 = τ c X , c X < τ c X , and for any other closure operator c on X with c 2 = c and c X < c, then τ c X < c as well.
Definition 2.35. We call τ c X the topological modification of c X In the next proposition, we see that topological modification of a closure structure has a characterization in terms of continuous maps. Let (X, c) be a closure space, and let c ′ be a closure operator on X. c ′ is the topological modification τ c of c iff the following two statements are satisfied: (1) c ′ is a topological closure operator (2) For any topological space (Y, τ ), a map f : X → Y is continuous as a map from (X, c) → (Y, τ ) iff f is continuous as a map from (X, c ′ ) → (Y, τ ).
In categorical language, this gives the following. Proof. On objects, we define τ (X, c) := (X, τ c). For each continuous function Since τ is associative, preserves composition, and preserves identity maps, we have that τ : Cl → Top is a functor.
To see that τ and ι are adjoints, we must show that there exists a natural bijection between Hom Top ((X, τ c), (Y, τ Y )) and Hom Cl ((X, c), (Y, τ Y )). However, this follows immediately from Proposition 2.36.

2.5.
Connectivity. In the discussion of covering spaces in Subsection 4.3, we will need the following definitions and results on connectivity of closure spaces.
is connected as a closure space with the subspace closure structure c A .
2.6. New structures from old. We end this section with the constructions of closure structures on subspaces, disjoint unions, products, and quotients, as well as some basic results.
Let (X α , c α ) α∈I be a collection of closure spaces, let α∈I X α be the Cartesian product of the underlying sets, and let π α : α∈I X α → (X α , c α ) be the projection mappings from α∈I X α to X α . For each x ∈ α∈I X α , let By Corollary 2.13, Part 2, there exists a unique closure structure c Π on α∈I X α such that V x is a subbase for each x ∈ α∈I X α . We define the product closure operator on α∈I X α to be the closure structure c Π .
(4) Let f : (X, c X ) → Y be an onto mapping from a closure space (X, c X ) to a set Y . For any V ⊂ Y , we define the quotient closure operator Neighborhoods in subspaces have the following useful property.
We state the following useful results about these constructions from [11].
We now shift our attention to results on products.
Then the collection of sets of the form π −1 a (U ), U ∈ U a is a local subbase for x in X. Similarly, if each U a is a local base at x a ∈ X a , then the collection of all sets of the form ∩ n i=1 π −1 ai (U ai ), where U ai ∈ U ai for each i ∈ {1, . . . , n} is a local base at x ∈ X.
Theorem 2.47 ( [11], Theorem 17.C.6). Let π α : Π α∈A X α → X α be the projections of a product space onto its coordinate spaces. Then the π α are continuous for each α ∈ A, where Π α∈A X α is endowed with the product closure structure c Π .
Moreover, c Π is the coarsest closure on Π α∈A X α which makes all of the π α continuous.
As in the topological case, we have Finally, in the following proposition we see that taking subspaces and taking products commute, which we will need when we develop homotopy.
where c ΠU and c ΠX are the product closure structures on Π α U α and Π α X α , respectively.
For quotients, we will need the following Proposition 2.50. Let p : (X, c X ) → Y be a surjective map from a closure space (X, c X ) to a set Y . Then the quotient closure structure c p on Y induced by p is the finest closure structure on Y which makes p continuous.
Proof. Suppose there exists a closure structure c on Y which is finer than c p and for which p : ). However, this implies that p is not continuous, a contradiction. Therefore, c p is the finest closure structure on Y which makes p continuous, as desired.
In the following proposition and example, we show that products in Top are also products in Cl, but that quotients in Top are not necessarily quotients in Cl. Proof. The proposition follows immediately from the fact that the inclusion functor ι : Top → Cl is a right-adjoint, and therefore preserves limits. An alternate proof is given in [11].
Example 2.52 ( [11], Introduction to Section 33.B). We give a simple example here to show that the quotient closure structure on the quotient of topological spaces need not be topological. Consider the four point space A = (1, 2, 3, 4), with the closure structure and note that c 2 = c, so (A, c) is a topological space. Now consider the mapping f : {1, 2, 3, 4} → {x 1 , x 2 , x 3 } given by By Proposition 4, the quotient closure structure c f is The structure c f is not topological, however, since

A family of closure structures on metric spaces
We now define the closure operators on extended pseudometric spaces which will be our main examples.
Lemma 3.3. For any r ≥ 0, c r defined above is a closure operator on the metric space (X, d X ), and c 0 is the topological closure operator on X for the topology induced by the metric.
Proof. The proof follows easily from the definitions. First, we see that c r (∅) = ∅ for all r. Next, we note that since To see the last statement, note that c 2 0 = c 0 , so c 0 is topological, and furthermore that x ∈ c 0 (A) ⇐⇒ d(x, A) = 0, so c 0 equals the topological closure structure induced by the metric.
Definition 3.4. For fixed q, r > 0, we say that a function between metric spaces is (q, r)-continuous if, for every ǫ > 0 and x ∈ X, there exists a δ(x) > 0 such that An (q, r)-continuous function with δ independent of x is called absolutely (q, r)continuous, and if q = r, we simply say that f is r-continuous.
We now show that (q, r)-continuity on metric spaces and continuity for maps between the associated closure spaces is equivalent.
be metric spaces with closure operators c q and c r , respectively. Then a map f : X → Y is continuous as a map between closure spaces (X, Conversely, let f be (q, r)-continuous, and suppose that x ∈ c q (A) for some A ⊂ X. We claim that f (x) ⊂ c q (f (A)). To see this, first choose an ǫ > 0 and let and therefore there exists a y ∈ A with d(x, y) ≤ q + δ(ǫ, x). It follows that d(f (x), f (y)) < r + ǫ. Since f (y) ∈ f (A) and ǫ is arbitrary, it follows that d(f (x), f (A)) ≤ r, and therefore f (x) ∈ c r (f (A)), and the claim is proved.
The above holds for any x ∈ A, so we have f (c q (A)) ⊆ c r (f (A)). Furthermore, since A is arbitrary, f is continuous, as desired.
We now give a number of elementary results which aid in developing intuition and which will simplify several proofs in the following section.
The identity map obeys the following Lemma 3.6. For any metric space (X, d X ), the identity map Id : -continuous if at least one of the following holds: Proof. Note first that, if q ≤ r, then c q < c r . Furthermore, if the diameter of (X, d X ) is less than r, then c r (A) = X for any nonempty A ⊂ X, and therefore c q < c r as well. The result now follows from Proposition 2.26.
We have the following rule for composition.
By Lemma 3.6, the identity map in the diagram is continuous, and by Proposition 2.25, the composition g • id • f = g • f is continuous, which proves the lemma.
Similarly, we have the following lemma.
Lemma 3.8. If f : X → Y is (q, r)-continuous and p ≤ q and r ≤ s, then f is (p, s)-continuous.
Proof. Consider the diagram s ) By Lemma 3.6, the identity maps above are continuous, and by Proposition 2.25, the composition is continuous, which proves the lemma.
for all x, y ∈ X. Therefore, for any ǫ > 0, we have Taking δ := ǫ K , the result follows from Proposition 3.5. The following corollary follows immediately from the above with K = 1, but is nonetheless worth stating separately.
is continuous as a map between closure spaces.
where r 0 > 0 is a constant independent of the points x and y. Then c r = c ′ r for any 0 ≤ r < r 0 , where c r and c ′ r are the closure structures associated to d and d ′ , respectively.
The same argument with the roles of c r and c ′ r reversed shows that Proof. The result follows directly from Proposition 3.5 since, for sufficiently small ǫ > 0, there does not exist a δ > 0 which would make f (r, s)-continuous at x or x ′ .
Proof. First consider the case r = 0. Since every point in G 0 is a connected component, every subset of G 0 is a collection of connected components as well, and the result is true in this case. Now let r > 0, suppose that A ⊂ X is open and that A is not the union of connected components. Then there is a point x ∈ A and a point y ∈ X\A such (x, y) ∈ E. This implies that d(x, y) ≤ r. However, c r (X\A) = X\A, and therefore, for any x ∈ A, d(x, X\A) > r, a contradiction. Therefore A is a union of connected components in G r , proving the result. Proof. Note first that, for any x ∈ X and any ǫ > 0, the ball is a neighborhood of x. Fix an ǫ > 0, and consider the interior cover V := {B x } x∈X of X. By Theorem 2.27, {f −1 (B x ) |∈ V} is an interior cover of K. We recall from Definition 2.23 that K is compact iff every interior cover of K has a finite (not necessarily interior) subcover. Therefore, there exist a finite number of points Proof. Let y ∈ g(c(A)) for some A ⊂ X. We must show that y ∈ c r+2s (g(A)). By definition, there exists an x ∈ c(A) such that g(x) = y, and by hypothesis Taking the infimum over x ′ ∈ A on both sides, we obtain as desired.

Homotopy theory on closure spaces
In this section, we present the construction of the fundamental groupoid and fundamental group of a closure space (X, c), and then proceed to prove a general form of the Seifert-Van Kampen theorem in this setting, using interior covers of (X, c). Finally, after briefly describing covering spaces in Cl, we calculate the fundamental group of the circle for a family of closure structures from Section 3.
We define homotopic maps and homotopy equivalence of spaces as in the topological setting.   We say that f is homotopic to g rel X ′ , denoted by f ≃ g rel X ′ , iff there exists a continuous map H : for all x ∈ X ′ , t ∈ I. If X ′ = ∅, we simply say that f is homotopic to g.
Suppose now that for a function f : Then we say that f is a homotopy equivalence between (X, A, c) and (Y, B, c ′ ), g is the homotopy inverse of f , and (X, A, c) and (Y, B, c ′ ) are homotopy equivalent. Remark 4.6. As in the topological case, homotopy and homotopy equivalence are equivalence relations. The proof follows from Proposition 4.2, and is identical to that in the topological case. Proof. The proof proceeds as for topological spaces (as in, for instance, [25]), with several additional verifications to confirm that the homotopies involved are continuous for the respective closure structures. Let Π(X, c) be as in the statement of the theorem.
We first show that the constant maps c x : I → (X, A), c x (t) = x, represent the identities in Π(X, c). Let Since f (1) = g(0), f ⋆ g is defined, and it is continuous by Proposition 4.2. Note, furthermore, that g(1) = f (0). We define the function H : , 1 H is well-defined by the conditions on the endpoints of f, g and h, and H is continuous by Corollary 2.32. Since H(0, t) = (f ⋆ g) ⋆ h(t) and We have now shown that Π(X, c) is a groupoid.
Definition 4.9. We call Π(X, c) the fundamental groupoid of (X, c), which we write as Π(X) when the structure c is unambiguous. Given a subset A ⊂ X, we let Π(X, A) denote the full subgroupoid of X whose objects are the points of A, which we call the fundamental groupoid of (X, A, c). We further define π 1 (X, * ) := Π(X, * ) op , which we call the fundamental group of (X, c). We will write the product in π 1 (X, * ) as [v][w] = [v⋆w], and we write π 1 (X) when the basepoint is understood.
Remark 4.10. We define π 1 (X, * ) as Π(X, * ) op and not as Π(X, * ) in order to make the product in the fundamental group agree with the classical definition, i.e. where the group product of classes in π 1 (X) is written in the same order as the ⋆ product on functions, although clearly Π(X, * ) and Π(X, * ) op are isomorphic groups.

4.2.
A Seifert-van Kampen Theorem. Using the above lemmas, we present a Seifert-van Kampen Theorem for the groupoids Π(X, X 0 ) using general interior covers, based on the proofs in [6,8,14]. We begin by proving an important lemma that will be used in many of our computations. an open cover of I n . Let λ be the Lebesgue number for U, and choose k ∈ N which satisfies 1 k < 2λ √ n . Decomposing I n into k n cubes whose side is length 1 k now gives the result.
where the maps a and b are determined by the inclusions If X 0 meets each path-component in the two-fold and three-fold intersections of distinct sets in U, then c is the coequalizer of a, b in the category of groupoids.
The proof of the theorem closely follows the proof given in [8] for topological spaces, with the addition that we use Lemma 4.11 to guarantee the existence of the required subdivisions of I and I 2 . We refer the reader to [7], Section 1.4 for a detailed discussion of the ideas in the proof in the case where U = {U 1 , U 2 }, as well as several illustrative diagrams of the homotopies involved.
We will need the following two lemmas. Proof. In Top, the homotopy extension property is equivalent to the existence of a retract r : X × I → X × {0} ∪ A × I. The existence of such a retract, however, is also sufficient to prove the homotopy extension property in Cl. That is, given a closure space (Z, c Z ) and continuous maps f : A × I → Z and g : X × {0} → Z such that f (a, 0) = g(a, 0) for a ∈ A, we define a map F : F is well-defined by the assumptions on f and g. Since A and X are Hausdorff and the inclusion A ֒→ X is a cofibration in Top, A must be closed in X, and therefore the collection {X × {0}, A × I} is a locally finite closed cover of X × I. Since the restrictions of F to each of the closed subsets A× I, X × {0} ⊂ X × I are continuous by definition, it follows from Corollary 2.32 that F is continuous.
Lemma 4.14. Let U = {U α } α∈A be an interior cover of a closure space (X, c), indexed by the set A. Let n ≥ 1, and suppose that f : I n → (X, c) is a map of a topological cube such that f maps the set I n 0 of vertices of I n to a set X 0 ⊂ X. Suppose, furthermore, that X 0 meets each path component of every k-fold intersection of distinct sets of U for any 2 ≤ k ≤ n + 1.
Then, I n may be subdivided into sub-cubes {c λ } λ∈B by planes parallel to x i = 0, i = {0, . . . , n}, such that, for every λ ∈ B there exists an α(λ) ∈ A with f (c λ ) ⊂ U α(λ) for some α ∈ A. In addition, there exists a map g : I n → (X, c) with f ≃ g rel Furthermore, the cubical subdivision can be taken to be a refinement of any prespecified cubical subdivision of the cube. In particular, we can arrange for the subdivision to refine pre-specified cubical subdivisions on the faces I n−1 × {0} and I n−1 × {1}.
Proof. Let U ′ := f −1 (U) and note that it is an interior cover of I n by Theorem 2.27. Since the Lebesgue covering dimension of I n is n, there exists an open refinement V of U ′ such that every point in I n meets at most n + 1 sets of V. By Corollary 4.11, there exists a decomposition of I n by planes parallel to x i = 0, i = {0, . . . , n} into sub-cubes c λ such that each c λ ⊂ V , for some V ∈ V. If I n−1 × {0} and I n−1 × {1} are already given subdivisions d ′ λ ′ and d ′′ λ ′′ , respectively, then we may further subdivide I n so that the restriction of the cubes c λ to I n−1 × {0} and I n−1 × {1} refines d ′ and d ′′ , respectively. Now let I (0) ⊂ I (1) ⊂ · · · ⊂ I (n) = I n be a cell decomposition of I n where I (0) is the union of the vertices of the sub-cubes c λ , and the cells in dimension k are the k-dimensional faces of the c λ . We proceed by induction on the dimension k of the skeleta of I n .
We first consider the case k = 0. Consider a point v ∈ I (0) , and let c λ1 , . . . , c λm be the sub-cubes that contain v.
If at least two of the U αi are distinct, then X 0 intersects each path component of U := ∩ m i=0 U αi . However, we also have that f (v) ∈ U , and therefore there exists a point x ∈ X 0 ∩ U and a path γ vx : I → X such that γ vx (I) ∈ U, γ(0) = v, γ(1) = x.
If v ∈ X 0 , then we take x = v and γ vx to be the constant map.
Suppose, on the other hand, that U αi = U αj for all i, j ∈ {0, . . . , m}. We call this set U . Let a : I → I n in I n be a path in I n with a(0) = v, a(1) = (0, . . . , 0). If f • a ⊂ U , then f • a is a path in U from f (v) to f (0, . . . , 0) ∈ X 0 , and we let γ = f • a.
If f • a ⊂ U , then let U ′ = ∪ U ′′ ∈U ,U ′′ =U U ′′ . Then U ′ = U by definition, and {U, U ′ } is an interior cover of X. By Corollary 4.11, there is a subdivision of I into intervals [t i , t i+1 ] such that each interval is mapped to either U or where U ′′ ∈ U, U ′′ = U . Since X 0 intersects every path component of every twofold intersection of distinct elements of U, it follows that there is a path b joining a point w ∈ X 0 ∩U ′′ ∩U to f • a(t i ). We now let γ be the concatenation of f • a([0, t i ]) and b, which, by construction, joins f (v) with a point in X 0 .
To complete the proof of the lemma, we define H(x, t) := H (n) (x, t) and g(x) := H(x, 1).
We now proceed with the proof of Theorem 4.12.
Proof of Theorem 4.12. We begin with a remark on notation. Throughout the proof, the product ⋆ will be reserved for the ⋆-product of two functions, * will indicate the composition of two functors between groupoids, and we will write • for the composition of two morphisms of a groupoid.
We need to show that, for any groupoid Γ and any functor F : such that F * a = F * b, then there exists a unique functor G : Π(X, X 0 ) → Γ such that G * c = F . We first remark that for any closure space (U, c U ) and U 0 ⊆ U , a path λ : ([a, b], {a, b}) → (U, U 0 ) may be taken to be a representative of the class [λ(tb + (1 − t)a)] ∈ Π(U, U 0 ) by reparametrizing the domain. Abusing notation slightly, we will use the parametrizations interchangeably in what follows, and, in particular, we will refer to homotopy classes of the form [λ(tb + (1 − t)a)] simply as [λ([a, b])].
Let λ : (I, ∂I) → (X, X 0 ) be a path in X with endpoints in X 0 , and suppose that there exists a decomposition of the I into subintervals [t i−1 , t i ], i ∈ {1, . . . , k}, where for every i, If a path λ admits such a decomposition, then we say that λ is admissible. Denote by λ i : [t i−1 , t i ] → X the restriction of λ to the interval [t i−1 , t i ]. Denote by [λ i ] ∈ Π(X, X 0 ) the class of λ i in the fundamental groupoid of (X, X 0 ). Since λ(t i ) ∈ X 0 for all i, λ i also represents a class in Π(U αi , U αi 0 ), which we denote by , and other functions from Π(U αi , U αi 0 ), when there is no ambiguity. Note that c[λ i ] αi = [λ i ], and that, by reparametrizing, we have [λ k ] • · · · • [λ 1 ] = [λ] ∈ Π(X, X 0 ). Therefore, if G exists, then for any admissible path λ, G must satisfy For a general path α : (I, ∂I) → (X, X 0 ), Lemma 4.14 with n = 1 gives a homotopy from α to an admissible path λ. Therefore, if F exists, then we must additionally have G[α] = G[λ]. Since G[λ] must be given by Equation 4.3, we conclude that, if G is well-defined, it must be unique.
To see that G is well-defined, it remains to show that, for two admissible homotopic paths λ, λ ′ : (I, ∂I) → (X, X 0 ) which are in the same homotopy class rel ∂I, we have are the subdivisions of I for λ and λ ′ , respectively. LetH : I 2 → (X, c) be a homotopy rel ∂I from λ to λ ′ . Since X 0 intersects every path component of every two-fold and three-fold intersection of distinct sets of U, we may apply Lemma 4.14 with n = 2 to obtain maps H : I 2 × I → (X, X 0 ) and g(s, t) := H(s, t, 1), and a subdivision of I 2 into squares {c ij } such that the conclusions of Lemma 4.14 are satisfied. In particular, the restriction of {c ij } to I ×{0} and I × {1} refines the subdivisions of I given by We first show that for the paths H(·, 0, 1) and λ, we have G  H(I, 0, 1)]. Using a succession of these homotopies, each inside the relevant U αi , we arrive at a homotopy between λ ⋆ H(1, 0, I) and g(·, 0) ⋆ H(I, 0, 1), and we also see that The proof that G[λ ′ ] = G[g(·, 1)] is identical. We must now show that G[g(·, 0)] = G[g(·, 1)]. The proof proceeds largely as above. We first note that, sinceH is a homotopy rel ∂I, λ(0) =H(0, ·) = λ ′ (0) ∈ X 0 and λ(1) =H(1, ·) = λ ′ (1) ∈ X 0 , and in particular both are constant functions. By the construction of H in Lemma 4.14, the functions H(ǫ, ǫ ′ , t) are constant as well, where ǫ, ǫ ′ ∈ {0, 1} Furthermore, H(s, 0, t) = H(r(s, 0, t)), where r is the retract of the face 0 × I × I onto 0 × I × 0 ∪ 0 × ∂I × I. Since H is constant on the latter set, it follows that H is constant on 0 × I × I. Similarly, H is constant on 1 × I × I, and therefore g(s, t) = H(s, t, 1) is a homotopy rel ∂I between H(·, 0, 1) and H(·, 1, 1). Now recall that, by Lemma 4.14, g(v 1 , v 2 ) ∈ X 0 for any vertex v = (v 1 , v 2 ) of a cube c ij . Therefore, for all i and j, the paths g( are homotopic in U αij . Denote by H ij a homotopy that connects them. Note that that the right side of Equation 4.3 is constant for two paths which are homotopic through such a homotopy H ij . Since g(0, t) and g(1, t) are constant, by concatenating a sequence of the homotopies H ij , we arrive at a homotopy between g(0, t) ⋆ g(s, 1) and g(s, 0) ⋆ g(1, t) which keeps Equation 4.3 constant. We therefore see that G depends only on the homotopy class of λ.
Since G is a functor by construction, this completes the proof.
As in the topological case, the traditional version of the Seifert-van Kampen theorem for groups is now a formal consequence of Theorem 4.12.
Theorem 4.15 (Seifert-van Kampen Theorem for Groups). Let (X, c X ) be a closure space with interior cover U := {U 1 , U 2 } be an interior cover. Let U 12 := U 1 ∩U 2 , and suppose that U 1 , U 2 , and U 12 are path-connected and the point * ∈ U 12 . Let i α : U 12 → U α and j α : U α → X denote the respective inclusions. Then is a pushout in the category of groups.
Proof. Apply Theorem 4.12 to the cover U with X 0 = * .

4.3.
Covering spaces and the fundamental group of (S 1 , c r ). In this section, we will compute the fundamental group of the circle endowed with different closure operators, as well as that of several graphs which may be seen as subspaces of S 1 with the induced closure operator. The main complication we encounter is that, depending on the closure structure, we are no longer guaranteed the existence of a lift to a contractible covering space p : E → S 1 for maps or homotopies of maps from S 1 to S 1 , and, when a lifting naively appears to exist, such a lifting starting at a given point x 0 ∈ E may no longer necessarily be unique. We illustrate these problems with the following examples.  Proof. Since p is a contraction of metric spaces, it is continuous as a map (R, c r ) → (S 1 , c r ) by Corollary 3.10.
The following two propositions demonstrate that there are genuinely new phenomena that must be accounted for in order to develop a theory of covering maps for closure spaces. In Proposition 4.19, we find lifts to what one expects to be a covering space that are not unique, and in the 4.21, we construct two homotopic maps which do have unique lifts to a candidate covering space, but where there exists a homotopy between them on the base space which does not have a lift. Proof. We will construct two non-equal lifts of γ which take the same value at 0. Define the functions f, g : [0, 1] → R by f (x) = 0, and is indiscrete, both f and g are continuous with respect to the closure structures, and that they both lift γ to R, i.e. pf = γ = pg, starting at the basepoint f (0) = g(0) = 0, so the lift of γ given an initial point in this case is not unique.
For the second example, we will require the following lemma.
Suppose that there exists a fixed 0 < α < 1 2 such that, for every x ∈ X, there is an open neighborhood U ⊂ X of x with f (U ) ⊂ I f (x),α := p((z − α, z + α)), where z ∈ p −1 (f (x)). (Note that the set I f (x),α is independent of the choice of z ∈ p −1 (f (x)).) Then, for any two continuous lifts F, G : X → (R, c 1/2 ) of f , there exists an integer n F,G ∈ Z such that F (x) − G(x) = n F,G for all x ∈ X.
The above examples illustrate several ways in which the standard methods for computing the fundamental group of S 1 in the topological category do not immediately generalize to closure structures on S 1 . We will solve these problems by incorporating the closure structure, via neighborhood systems, explicitly into the definitions of a covering map, analogously to how the topology of a space appears in the standard definition of a covering map for topological spaces. Once the new definitions are in place, we will see that the familiar construction of the fundamental group of S 1 then generalizes to this setting.
We begin with a brief discussion of covering maps in closure spaces. (F, c F ) is another closure space, let q 1 : U × F → U denote the projection onto the first factor. We say that a homeomorphism φ : p −1 (U ) → U × F such that q 1 • φ = p is a trivialization of p over U . When such a φ exists, p is said to be trivial over U. We say that p is locally trivial if there exists an interior cover U of B such that p is trivial over U for each U ∈ U.

every point is both open and closed)
, and there is an interior cover U B for B such that p is trivial over U for every U ∈ U B . We call F b the fiber of p at b.
Remark 4.24. We observe that, for a covering p, since is homeomorphic to a disjoint union of copies of U embedded in E. The summands are called the sheets of the covering p over U .
We now give a proposition, which gives sufficient conditions for the uniqueness of a lift. (1) Wb ∩ Wb ′ = ∅ for anyb =b ′ ,b,b ′ ∈ p −1 (b), and (2) The restriction of p to each Wb is injective into U b . Then the lift g of f is unique.
Proof. Let g, g ′ : (X, c X , * ) → (E, c E , * ) be lifts of f with g( * ) = g ′ ( * ). Let x ∈ X, b = g(x) ∈ E andb ′ = g ′ (x) ∈ E so that p(b) = p(b ′ ) = f (x), and consider the sets Wb and Wb ′ . Since Wb and Wb ′ are neighborhoods ofb andb ′ , respectively, it follows that N = g −1 (Wb) ∩ g ′−1 (Wb ′ ) is a neighborhood of x ∈ X. If g(x) = g ′ (x) then g|N = g ′ |N , since every point in Wb and Wb ′ is the pre-image of at most one It follows that the sets {x ∈ X | g(x) = g ′ (x)} and {x ∈ X | g(x) = g ′ (x)} are both open and closed, and therefore they are either equal to X or ∅, since X is connected. Since g( * ) = g ′ ( * ), we have that X = {x ∈ X | g(x) = g ′ (x)}, and therefore g = g ′ as desired.
In the next propositions, we give severeal examples of covering maps.
Proof. Let 0 ≤ r < 1 3 . Let x ∈ [0, 1), so p(x) = [x] ∈ S 1 . Define the set U x := {[y] ∈ S 1 | x − 1 3 < y < x + 1 3 }, and note that U x is a neighborhood of [x]. For every n ∈ Z, define x , c r ) ∼ = (U x , c r ) are homeomorphic as closure spaces, and, as a set, Furthermore, the inclusion map i : x , c r is a colimit in Cl, and i is the map guaranteed by the universal property of colimits. Since, for any n = n ′ , we have d(U n x , U n ′ x ) ≥ 1 3 , it follows that i −1 is continuous as well, and therefore we have the closure space homeomorphism as closure spaces, where Z is given the discrete closure structure and c Π is the product closure structure, it follows that p is a covering map. Now consider the case r ≥ 1 3 . By Theorem 2.42, a closure space (X, c X ) is connected iff it is not the union of two nonempty disjoint open subsets, or, equivalently, that there is no proper subset of X that is both open and closed. Now note that, with r ≥ 1 3 , if U is a neighborhood of a point x ∈ S 1 , then p −1 (U ) is connected with respect to the subspace convergence structure. However, each sheet of U × p −1 (x) is both open and closed, and therefore U × p −1 (x) is not connected. Therefore, U × p −1 (x) and p −1 (U ) cannot be homeomorphic, and p is not a covering map.
The closure structure on X × B E, denoted c ×B , is given by the subset closure structure.
be a closure space, and suppose that g : Y → X and h : Y → E are continuous and make the diagram Then there is a map φ : Y → X × B E of sets given by φ(y) = (g(y), h(y)).
Since g and h are continuous, φ is continuous as a map to the product space (X × E, c X×E ), i.e. as a map (Y, c Y ) → (X × E, c X×E ). However, since the range of φ is contained in X × B E, it follows that, for any Y ⊂ X, we have Therefore, φ is continuous as a map from (Y, c Y ) → (X × B E, c ×B ), and the result follows.
Theorem 4.29. The pullback of a covering map is a covering map.
Proof. Let p : (E, c E ) → (B, c B ) be a covering map, suppose the map f : (X, c X ) → (B, c B ) is continuous, and let (X × B E, c ×B ) be the pullback of the resulting diagram, and denote by q : (X × B E, c ×B ) → (X, c X ) the resulting projection. Let b = f (x) for some x ∈ X. Since p is a covering map, there exists a neighborhood  Proof. We first note that ι is well defined, since for any two representatives of [k], we have ι[k] = k n = k n + a = ι[k + an], for any a ∈ Z. Next, since for any point [k] ∈ Z n , ι(c m [k]) ⊂ c m/n (ι[k]) by definition, it follows that ι is continuous.
Let (X, c X ) be a closure space, and suppose that f : (X, c X ) → (R, c m/n ) and g : (X, c X ) → (Z n , c m ) are continuous maps such that the diagram is an integer, it follows that f (x) = k + g(x) n for some integer k. Therefore, n · f (x) is an integer for all x ∈ X, so φ is welldefined, and, furthermore, f =ι • φ and g = q • φ, whereι is the continuous map ι(k) = k n ∈ R, and q is as in the hypothesis. Furtherore, since f is continuous, we have, for any A ⊂ X, that f (c X (A)) ⊆ c m/n (f (A)). However, this implies that φ(c X (A)) = n · f (c X (A)) ⊆ (n · c m/n (f (A))) ∩ Z = c m (n · f (A)) ∩ Z = c m (φ(A)), and therefore φ is continuous. It follows that (Z, c m ) is the pullback of p along ι. By the unicity of pullbacks, we further have (Z, c M ) ∼ = (Z × S 1 R, c × S 1 ).
Corollary 4.32. The map q : (Z, c m ) → (Z n , c m ) in Proposition 4.31 above is a covering map iff 0 ≤ m < n 3 . Proof. By Proposition 4.31, the map q is the pullback of the covering map p : (R, c m/n ) → (S 1 , c m/n ). By 4.26, p is a covering map iff 0 ≤ m n < 1 3 . By Theorem 4.29, q is therefore a covering map if 0 ≤ m < n 3 . Conversely, if m ≥ n 3 , the space q −1 (U ) is path connected for any neighborhood U of any point [k] ∈ Z n , and therefore q is not a covering map.
The remainder of the calculation of the fundamental group of (S 1 , c r ) for 0 < r < 1 3 now reduces to a generalization of the classical calculation of the fundamental group of S 1 . The calculation for the fundamental group of (Z n , c m ) will proceed similarly, although clearly this case is specific to the closure space setting. We present the calculations here as corollaries of the theorem that, given a covering map p : E → B, where E is a simply connected space, π 1 (X, * ) is isomorphic to the automorphism group of the covering. The proof is nearly verbatim the classical proof for topological spaces, with a few delicate points regarding when to appeal to open sets of a closure structure and when to use neighborhoods. We have adapted our treatment from [26] and [25], and we include several auxiliary results on covering spaces. We begin with the following preparatory results.
Proof. Since p is a covering map, the hypotheses of Proposition 4.25 are satisfied. The result follows. Lemma 4.35. Let U be an an interior cover of X × I with the product closure structure. For each x ∈ X there exists a neighborhood V (x) of x ∈ X and n = n(x) ∈ N such that, for 0 ≤ i < n, the set V (x) × i n , (i+1) n is contained in some member of U.
Proof. By Theorem 2.46, if U is a local subbase for x ∈ X, and V is a local subbase for t ∈ [0, 1], then the family {π −1 1 (U ) | U ∈ U} ∪ {π −1 2 (V ) | V ∈ V} is a local subbase for (x, t) ∈ X ×I, where the π i , i ∈ {1, 2} are the projections onto X and I, respectively. It follows that every neighborhood N t ⊂ X ×I of a point (x, t) ∈ X ×I contains a neighborhood of (x, t) of the form V (x,t) = U t (x)× (t 1 , t 2 ) ⊂ X × I, where U t (x) is a neighborhood of x ∈ X and (t 1 , t 2 ) is a neighborhood of t ∈ [0, 1]. In particular, for every (x, t), there is a neighborhood of the form of V (x,t) contained in any set of N which is a neighborhood of (x, t). Since I is compact, a finite number of such neighborhoods V (x,t) cover {x} × [0, 1]. Let {t 0 , . . . , t k } be the points in I such that the family {V (x,ti) } k i=0 covers {x} × I, and let λ be the Lebesgue number of the cover {π 2 (V (x,ti) )} k i=0 of I given by the projection onto I. Choosing V (x) := ∩ k i=1 U ti (x) and n > 1 λ , the lemma follows.
We now give the definition of a fibration in Cl for use in what follows.   Proof. Let h : (X, c X ) × I → (B, c B ) and a : (X, c X ) → (E, c E ) be a homotopy and initial condition, respectively, i.e. pa(x) = hi 0 (x), where i 0 : X → X × I is the map x → (x, 0). Since p is a covering map, there is a neighborhood system U B on B such that for each U ∈ U B , p −1 (U ) = U × F b for some b ∈ U . Since h is continuous, V := h −1 (U) is a neighborhood system on X × I. Consider a point x ∈ X, and let V (x) ⊂ X and n x ∈ N be the neighborhood of X and the natural number guaranteed by Lemma 4.35, respectively, so that We proceed by induction on i. First consider the case i = 0. Since p : p −1 (U ) → U is a projection, it is a fibration by 4.37, and therefore h : Since the lifting of h(y × I) is unique, The maps H i and nx is continuous by 2.32 as a map from the subspace V (x)×[0, i+1 nx ] ⊂ (X, c X ) with the induced closure operator. I is connected, Proposition 4.34 shows that the lifting of h|x × I to E is unique for any x, and therefore the liftings on each V (x) × I combine to a well-defined lifting H : X × I → E of h. Since the V (x) × I form an interior cover of (X, c X ) × I, we have that H is continuous by 2.33. Since (X, c X ) was arbitrary, p is a fibration.
The remainder of the results required for our calculations are primarily algebraic, and follow verbatim the discussion in [26], Section 3.2, replacing topological spaces with closure spaces. We give a full exposition here for convenience, as well as to establish notation and definitions for the computations which follow. Proof. We first recall that there exists a homeomorphism φ : (I n × I, I n × {0}) → (I n × I, (I n × 0) ∪ (∂I n × I)).
Using this homeomorphism, we may transform the above lifting problem into a homotopy lifting problem for the cube I n . Since p has the homotopy lifting property for cubes, a function H : I n × I exists with the desired properties.  at (B, b) iff im(f ) = ker g, where ker g := g −1 (c). For a group, we take the basepoint to be the identity, and for the set π 0 (X) for some closure space (X, c), we write π 0 (X, a), to indicate that the basepoint is [a].  −1 (b). Then there exists a map ∂ x : π 1 (B, p(x)) → π 0 (F b , x) making the sequence Furthermore, the preimages of elements under ∂ x , i.e. ∂ −1 , are the left cosets of π 1 (B, b) with respect to p * π 1 (E, x). Finally, the preimages of π 0 (i) = i * : π 0 (F b , x) → π 0 (E, x) are the orbits of the π 1 (B, b)-action on π 0 (F b , x).
For each such path we will define a map v # : π 0 (F b ) → π 0 (F c ). Let x ∈ F b . Since p has the homotopy lifting property for I, we may find a lifting V : We claim that v # is well-defined and depends only on the class [v * ] ∈ Π(B). Let Then there is a path γ : I → F b which connects x to x ′ . Let v ′ : I → B be a path from b to c with v ≃ v ′ rel ∂I, and let h : I × I → B be the homotopy from v to v ′ . Finally, let V, V ′ : I → E be liftings of v and v ′ to E with initial points γ(0) and γ(1), respectively. The maps γ, V , and V ′ give a continuous map a : I × ∂I ∪ 0 × I → E, defined by a(s, 0) = V (s), a(s, 1) = V ′ (s), a(0, t) = γ(t). We also have that p • a = h • i, where i : I × ∂I ∪ 0 × I → I × I is the inclusion. Then Lemma 4.39 gives a lifting H of h with initial condition a.
. This shows that the map v # is well defined and depends only on the class [v] ∈ Π(X), proving the claim. Additionally, it follows from the definitions that . We use this to define a map ∂ x : With this definition, we turn our attention to the exactness of the sequence in the theorem. By the definitions of the maps, we see that the composition of any two of them is equal to the basepoint. We now check that the kernel of every map is contained in the image of the previous one.
Let [u] ∈ ker p * ⊂ π 1 (E, x) and let h : I × I → B be a null-homotopy of p • u. Let a : I × 0 ∪ ∂I × I → E be the map a(s, 0) = u(s) and a(ǫ, t) = x, ǫ = 0, 1, and consider the lifing problem for h with initial condition a. The lifting H of h is a homotopy of loops beginning at u and ending at a loop u ′ with p • u ′ = b, which implies that [u ′ ] is in the image of i * .
Let We have now shown that the sequence in the theorem is exact, and we proceed to prove the final statements. 1 (B, b), and choose liftings U ,V of u and v, respectively, with is exact and i * is surjective. Therefore, E is path connected iff π 1 (B, b) acts transitively on F b . The isotropy group of x ∈ F b is the image of p * : π 1 (E, x) → π 1 (B, b).
The result now follows from Theorem 4.41.
Definition 4.45. Let p : (E, c e ) → (B, c B ) be a covering. A right G-action E × G → E, (e, g) → eg is said to be totally discontinuous iff every x ∈ E has a neighborhood U with U g ∩ U = ∅ and U g ∪ U is not connected for all g ∈ G. We call p a G-principal covering if G is totally discontinuous and p(xg) = p(x) for all g ∈ G and such that the induced action on each fiber  B, c B ) be a right G-principal covering with path-connected total space (E, c E ). Then the sequence of groups and homomorphisms . The image of p * is a normal subgroup. The space E is simply connected iff δ x is an isomorphism. Thus, if E is simply connected, G is isomorphic to the fundamental group of B.
Proof. We first recall the right action of where V is a map which lifts v and has initial point x. By Proposition 4.44, the point V (1) only depends on the homotopy class of [v]. Now let g ∈ G, and observe that xg x. It follows that V ′ g is a lift of v which begins at gx, and by Proposition 4.34, the uniqueness of liftings for covering maps, we have that W = V ′ g. Therefore, the actions of G and π 1 (B, b) on F b commute. Fix x ∈ E. Since G is free and transitive on F b , for every [v] ∈ π 1 (B, b) and ). This defines a map Γ x : To see that it is a homomorphism, we compute Define a map ρ x : G → F b by ρ x (g) = x · g −1 . Since G acts freely and transitively, ρ is a bijection. Furthermore ρ 1 (B, b).
Since st + r < st + r + ǫ, there exist δ ′ , β > 0 such that st + r + δ ′ t + (s + r + δ)β < st + r + ǫ Similarly, we may choose α and δ such that Choosing α, δ > 0 to satisfy both of these constraints, we conclude that H is continuous at (s, t). (Note that this argument also covers t ∈ {0, 1}.) Since (s, t) ∈ R × I was abitrary, H is therefore continuous. Since H is a homotopy from (R, c r ) to the constant map, we conclude that (R, c r ) is contractible.
Theorem 4.48. For 0 ≤ r < 1 3 , π 1 (S 1 , c r ) ∼ = Z. Proof. For r < 1 3 , the map p : (R, c r ) → (S 1 , c r ) is a covering map by Proposition 4.26, and the map R × Z → R, (n, t) → t + n defines a totally discontinuous, right Z-action R such that p(t · n) = p(t) for any n ∈ Z. Since the induced action on each fiber is transitive, p is a right Z-principal covering map. Since (R, c r ) is contractible, the conclusion follows from Theorem 4.46.
We now compute π 1 (Z n , c m ) for 1 ≤ m < n 3 . The proof is similar to the one above for (S 1 , c r ). We begin, as above, by proving the contractibility of the total spaces of the maps q : (Z, c m ) → (Z n , c m ). Proof. We first remark that a retract of a contractible space is contractible. Indeed, if the inclusion ι : (U, c U ) → (X, c X ) has a left inverse r : (X, c X ) → (U, c U ) and (X, c X ) is contractible, then the composition is a contraction of U if H is a contraction of X.
Corollary 4.50. For 1 ≤ m < n 3 , π 1 (Z n , c m ) ∼ = Z. Proof. For 1 ≤ m < n 3 , the map p : (Z, c m ) → (Z n , c m ) is a covering map by Proposition 4.32, and the map Z × Z → Z, (k, j) → k + j defines a totally discontinuous, right Z-action such that p(k · j) = p(k + j) = p(k) for any n ∈ Z. Since the induced action on each fiber is also transitive, p is a right Z-principal covering map. Since The result now follows from the five-lemma.
Using a homotopy equivalence directly, we have, for r ≥ 1 2 , Theorem 4.52. π 1 (S 1 , c r ) = {1} for r ≥ 1 2 Proof. If r ≥ 1 2 , then the closure structure c r is indiscrete, i.e. c r (A) = S 1 for any A ⊂ S 1 . Any function to (S 1 , c r ) is continuous in this case, and therefore (S 1 , c r ) is contractible.
Similarly,  Where F a,b denotes the free group on two generators.
For r i ≥ 1 2 , i = 1, 2, the closure structure c X is indiscrete. Therefore any map to X is continuous, and X is contractible.

4.4.
Combinatorial homotopy on graphs and simplicial complexes. In this section, we briefly show how to put a closure structure on graphs and, more generally, on k-skeleta of simplicial complexes.
Suppose X is an abstract simplicial complex. Denote by X (k) the k-skeleton of X, S(X) the set of simplices of X, and S (k) (X) the set of k-simplices of X. Let P(S (k) (X) be the powerset of the set of k-simplices of X. We define an operator c X,k : P(S (k) (X)) → P(S (k) (X)) by (4.5) c X,k (σ) = σ ∪ {γ ∈ S (k) (X) | γ ∩ σ ∈ S (k−1) (X) or γ ∪ σ ∈ S (k+1) (X)}, for σ ∈ S (k) (X). Now let c X,k (A) := σ∈A c X,k (σ), for A ∈ P(S(X)). Similarly, for the vertices V in a directed graph G = (V, E) we may define (4.6) c E (U ) = {v ∈ V | v ∈ U or (w, v) ∈ E for some w ∈ U } We then have Proposition 4.55. c X,k and c E are Čech closure operators.
Proof. Immediate from the definitions of Čech closure operators, c X,k , and c E .
For 'circular graphs' on three vertices, we have the following. Proof. The closure structure is indiscrete, which implies that any map to V is continuous. Therefore V is contractible.
As the spaces (Z n , c n,m ) can be interpreted as graphs by defining V = Z n , E := {(i, j) | j ∈ c(i), the calculations in Subsection 4.3 give a number of calculations of fundamental groups for graphs with the induced closure structure.
Remark 4.57. Applying the above homotopy theory to the closure structures given by the c X,k for a simplicial complex X results in homotopy groups in a similar spirit to those found in [1][2][3][4]. Additionally, the homotopy theory for (V, c E ) closely resembles that given in [20]. It is not yet clear whether the homotopy groups developed here are the isomorphic to those in [1][2][3][4] for graphs and simplicial complexes, or to those in [20] for directed graphs. 4.5. Persistent homotopy. We remark here that the closure structures given in Section 3 can be used to define so-called 'persistent' homotopy groups in the following way. If q ≤ r, then the identity maps (4.7) ι (q,r) : (X, c q ) → (X, c r ) are continuous, and therefore induce maps ι (q,r) n : π n (X, c q ) → π n (X, c r ), n ≥ 0, which are homomorphisms for n ≥ 1. We define the persistent homotopy groups of a metric space X by π (q,r) n (X, d) = im ι (q,r) n , n ≥ 1 and the persistent components of X by π (q,r) 0 (X, d) = im ι (q,r) 0 .