On U-equivalence spaces

In this paper induced U-equivalence spaces are introduced and discussed. Also the notion of U-equivalently open subsets of a U-equivalence space and U-equivalently open functions are studied. Finally, equivalently uniformisable topological spaces are considered.


Introduction
Non-Archimedean spaces have been considered from different viewpoints [1,2,3,4,5,11,13,14,15]. In this paper we deal with U-equivalence spaces. These spaces have been introduced first in 2014 [12]. A U-equivalence space (X, U) is a set X along with a collection U of equivalence relations on X such that U is closed under finite intersections. A U-equivalence space is a structure close to uniform spaces [7,8,10] and fuzzy uniform spaces [6,9]. A function f : X −→ Y where (X, U) and (Y, V) are two U-equivalence spaces is called Moreover if (X, U) is a U-equivalence space, then the collection T U = {G ⊆ X| for each x ∈ G, there exists U ∈ U such that U[x] ⊆ G} is a topology on X with the base {U[x]|U ∈ U, x ∈ X} where U[x] = {y ∈ X|(x, y) ∈ U}. We refer to T U as the U-induced topology. We are going to consider induced U-equivalence classes and U-products. Also we will introduce and discuss U-equivalently open subspaces, and equivalently uniformisable spaces.
2 Induced U-equivalence spaces and U-products Let {Φ i : X −→ X i } i∈I be an indexed family of functions where X is a set and for each i ∈ I, (X i , U i ) is a U-equivalence space. The idea is to induce a U-equivalence class [12] on X for which each Φ i is U-equivalently continuous without making the U-equivalence class on X unnecessarily strong.
Definition 2.1. Let S be a family of equivalence relations on a set X. Then the collection of all finite intersections of members of S (that forms a U-equivalence class on X) called the U-equivalence class generated by S and it is denoted by S .
Note that S is the smallest U-equivalence class on X which contains S. Proposition 2.2. Let {ϕ i : X −→ X i |i ∈ I} be an indexed family of functions where X is a set and for each i ∈ I, (X i , U i ) is a U-equivalence space. Then there exists a smallest U-equivalence class on X for which each Φ i is U-equivalently continuous. Proof: is U-equivalently continuous w.r.t. U and U is closed under finite intersections, then U ∈ U. Hence U ← ⊆ U.
The U-equivalence class U ← in the last proposition is called the induced U-equivalence class. Note that if each U i has a base B i , then the collection The following property is a characteristic of the induced U-equivalence classes. Proposition 2.4. Let Φ : X −→ Y and Ψ : Y −→ Z be functions where (X, U), (Y, Z) and (Z, W) are U-equivalence spaces. If Y has the induced U-equivalence class, then Φ is U-equivalently continuous if and only if ΨΦ is U-equivalently continuous.
Proof: The proposition follows from the equality where W runs through the members of W and the fact that V = W ← . Let (X, U) be a U-equivalence space and let A ⊆ X. By Corollary 2.3. (let Φ to be inclusion map) the collection {A × A ∩ U|U ∈ U} is the induced U-equivalence class on A that is called relative U-equivalence class and denoted by U/A (see [12]).
Hence U/A is the smallest U-equivalence class on A makes inclusion map U-equivalently continuous.
Definition 2.5. (a) Let (X, U) and (Y, V) be two U-equivalence spaces. A bijection Φ : X −→ Y is said to be a U-equivalence (function) if Φ and Φ −1 are U-equivalently continuous. (b) A function f : X −→ Y is said to be U-embedding if it is one to one and a U-equivalence when regarded as a function from (X, U) on to (f (X), V/f (X)).
Theorem 2.6. Let (X, U) and (Y, V) be two U-equivalence spaces and let Φ : X −→ Y be a function. Then the following statements are equivalent: (a) Φ is a U-embedding. (b) Φ is one to one, U-equivalently continuous and U = V ← Proof: (a)⇒(b). Since Φ is U-equivalently continuous when regarded as a function from (X, U), To complete the proof, we will prove that Φ −1 : Not all U-equivalently continuous injections are U-embedding's. For example let X be a set and U be the collection of all equivalence relations on X and V consists of X 2 and let Φ be the identity map on X. Then Φ is a U-equivalently injection when regarded as a function from (X, U) on to (X, V) but not U-embedding if |X| > 1. A useful sufficient condition is Proposition 2.7. Let Φ : X −→ Y be a U-equivalently continuous function where (X, U) and (Y, V) are U-equivalence spaces. Suppose Φ admits a U-equivalently left inverse. Then Φ is a U-embedding. Proof: The last equality is true because ΨoΦ is the identity map on X. Hence U = V ← . Proposition 2.8. Let {X j } be a finite covering of the U-equivalence space (X, U).
Suppose that X j is totally bounded [12] for each index j. Then X is totally bounded. Proof: We recall that X is totally bounded if for each U ∈ U there exists The converse of Proposition 2.8 is true. In fact we have Proposition 2.9. Let Φ : X −→ Y be a function where X is a set and (Y, V) is a U-equivalence space. If Y is totally bounded, then so is X with the induced U-equivalence class. Proof: The notation of induced U-equivalence classes is the standard way of giving a Uequivalence class to the cartesian product More precisely, let {(X i , U i )|i ∈} be given. For each i ∈ I let π i be the i-th canonical projection. By Proposition 2.2, there exists a smallest U-equivalence class Proof: The proof is an easy consequence of the equality According to [12] a U-equivalence space (X, U) is separated if the intersection of all members of U coincides with ∆X = {(x, x)|x ∈ X}.
Proposition 2.11. Let {(X i , U i )|i ∈ I} be a family of separated U-equivalence spaces. Then the U-product is separated .
Proof: Let x = (x i ) and y = (y i ) be two members of X = ΠX i and (x, Consequently the intersection of all members of ΠU i coincides with ∆X. Hence (ΠX i , ΠU i ) is separated.
Proof: Let j ∈ I be fixed and let X = ΠX i , We contend π j : X −→ X j is continuous when regarded as a function from topological space (X, This proves π j is continuous. Hence by definition of ΠT U i , ΠT U i ⊆ T ΠU i . For other way inclusion, suppose G ∈ T ΠU i and x ∈ G. Then there

U-Equivalently Open Subspaces
Among the subspaces of a U-equivalence space (X, U) special attention should be given to those which are U-equivalently open, in the sense that the inclusion map is Uequivalently open. We recall that a function f : for all x ∈ X. Roughly speaking, every U-equivalently open function is locally surjective (see [12]). For example Φ is always U-equivalently open when V is discrete i.e. V is the collection of all equivalence relations on Y .
Let (X, U) be a U-equivalence space. Then U is called rich if X 2 ∈ U. Proposition 3.1. Let (X, U) be a rich U-equivalence space and let A ⊆ X. Then the following statements are equivalent: Suppose α, β : X −→ Y are maps from X into Y . The coincidence set of α and β is the set C(α, β) = {x ∈ X|α(x) = β(x)}. Also if ϕ : X −→ Y is a function from the U-equivalence space (X, U) into the set Y , let us say that ϕ is transverse to X (or (X, U)) if (ϕ × ϕ) −1 (∆Y ) ∩ U = ∆X for some U ∈ U. Roughly speaking, ϕ is transverse to X if ϕ is one to one on a region of X 2 .
Proposition 3.2. Let (X, U) , (Y, V) and (Z, W) be U-equivalence spaces, α, β : X −→ Y be two U-equivalently continuous functions and ϕ : Y −→ Z be transverse to Y . If ϕoα = ϕoβ and U is rich then the coincidence set C(α, β) is U-equivalently open in X.
It is easy to see that U 0 [x] ⊆ C(α, β) for all x ∈ C(α, β) where Obviously any surjection from X onto the U-equivalence space (Y, V) is a Uequivalently surjection.
On the other hand if X = Y = R (the set of real numbers) and V = {R 2 }, we define f : X −→ Y by f (x) = x 2 , then f is a U-equivalently surjection but not a surjection.
Proposition 3.4. Let f : X −→ Y be a U-equivalently surjection. If f is Uequivalently open and U is rich, then f is a surjection. Proof: Hence y ∈ f (X) that means f is a surjection.
Definition 3.5. Let (X, U) be a U-equivalence space. A subset D of X is said to be U-equivalently dense (in X), if for each x ∈ X and each U ∈ U, there exists a ∈ D such that (a, x) ∈ U.
Note that D is U-equivalently dense iff D is a dense subset of topological space (X, T U ). Also, note that D is U-equivalently dense in X iff the inclusion map i : D → X is a U-equivalently surjection.
Proposition 3.6. Let (X, U) be a rich U-equivalence space and let A be a Uequivalently open and U-equivalently dense subset of X. then A equals X.
Proof: Since A is U-equivalently open, then the inclusion map i : A −→ X is Uequivalently open. If x ∈ X and U ∈ U, then there exists a ∈ A such that (a, x) ∈ U. So x ∈ U[i(a)]. Hence the inclusion map is also a U-equivalently surjection. Now Proposition 3.4 implies the inclusion map is a surjection. Hence A equals X.
We say the U-equivalence space (X, U) is connected if the topological space (X, T U ) is connected. We have the following strange result: Proposition 3.8. Every non-empty subset of a connected U-equivalence space is U-equivalently dense.
Proof: Let A be a non-empty subset of a connected U-equivalence space (X, U) and let U ∈ U. Since A is non-empty, then so is U The last statement is true because U is an equivalence relation on X. So U[A] is closed. Hence the conectedness of the U-equivalence space (X, U) implies that U[A] = X. Now if x ∈ X, then x ∈ U[A]. Thus (a, x) ∈ U for some a ∈ A that means A is U-equivalently dense in X. Corollary 3.9. Every U-equivalently open subset of a connected U-equivalence space is either empty or full. Proof: immediate from (3.6) and (3.8). Corollary 3.10. Let (X, U), (Y, V) and (Z, W) be U-equivalence spaces and let α, β : X −→ Y be U-equivalently continuous and ϕ : Y −→ Z be transverse to Y . If ϕoα = ϕoβ, U is rich and α(x 0 ) = β(x 0 ) for some point x 0 in X and if (X, U) is connected, then α = β.

Equivalently uniformisable spaces
In this section we consider topological spaces which are equivalently uniformisable. More precisely, let (X, T ) be a topological space. The question is: under what condition(s) there exists a U-equivalence class U on X such that T U = T . We begin with the following definition.
Definition 4.1. A topological space (X, T ) is said to be equivalently uniformisable if there exists a U-equivalence class on X such that T = T U .
Proposition 4.2. Let (X, T ) and (Y, V) be respectively topological and U-equivalence spaces and let f : X −→ Y be a topological equivalence from topological space (X, T ) on to topological space (Y, T V ). Then the topological space (X, T ) is equivalently uniformisable.
Proof: Let U be the induced U-equivalence class on X by V (i.e. U = We contend that T = T U . Suppose G ∈ T and x ∈ G. Since f is a topological equivalence, then f (G) ∈ T V . Hence there exists V ∈ V such that Conversely let G ∈ T U and x ∈ G. Since x ∈ G and G ∈ T U , then there exists U ∈ U Then G ′ ∈ T and x ∈ G ′ ⊆ G. This shows that T U ⊆ T . Consequently T = T U . This means that topological space (X, T ) is equivalently uniformisable. Proposition 4.3. Let (X, U) be a U-equivalence space and let Y ⊆ X. Then We now turn to the proof. We first show that T U /Y ⊆ T U /Y . Let G ∈ T U /Y and let y ∈ G. Then there exists V y ∈ U/Y and U y ∈ U This shows that T U /Y ⊆ T U /Y . Hence these two topologies are the same. Proof. The function f : X −→ Z is a topological equivalence where Z = f (X) regarded as a topological space with T V /Z (the induced topology on Z by T V ). But by Proposition 4.3, T V /Z = T V/Z . Hence by Proposition 4.2, (X, T ) is equivalently uniformisable.
By Proposition 4.4, a topological space is equivalently uniformisable if it can be embedded into a topological space whose topology induced by a U-equivalence class.
Definition 4.5. Let (X, d) be a pseudo metric space and let r be a positive real number. Then (a) d is called r-transitive if d(x, y) < r, d(y, z) < r implies d(x, z) < r for all points x, y and z in X.
(b) d is called transitive if for each r > 0, d is r-transitive.
For an example of r-transitive pseudo-metric, let X be a non-empty set and let where α is a positive real number. Then for each r > α, d α is r-transitive. Let d be a transitive pseudo-metric on a set X and let B d (r) = {(x, y) ∈ X × X|d(x, y) < r}.
Then the collection {B d (r) : r > 0} forms a U-equivalence class on X that is called the U-equivalence class generated by d and denoted by U d . Theorem 4.6. If a topological space can be embedded into a product of transitive pseudo-metric spaces, then it is equivalently uniformisable. Definition 4.7. Let D be a family of transitive pseudo metrics on set X and let S D be the collection {B d (r) : r > 0, d ∈ D}. The U-equivalence class generated by S D is called the U-equivalence class generated by D and denoted by U D . Note that since for each d ∈ D and each r > 0, d is r-transitive, then B d (r) is an equivalence relation on X.
Proposition 4.8. Let D be a collection of transitive pseudo metrics on a set X. Then T D = T U D where T D is the topology generated by sub-basẽ Proof: Let G ∈ T U D and let x ∈ G. Then U[x] ⊆ G for some U ∈ U D . By the Consequently T D = T U D .
Proposition 4.9. Let D be a collection of transitive pseudo metrics on a set X. For each d ∈ D, let X d be a copy of the set X and let Y = Π d∈D X d . Then the evaluation function f : (X, U D ) −→ (Y, ΠU d ) defined by f (x)(d) = x for all d ∈ D, is a U-embedding of (X, U D ) into (Y, ΠU d ).
Proof: Obviously f is an injection. Let U = (π d × π d ) −1 (U d ), U d = B d (r) for some positive real r. Then (f × f ) −1 (U) = U d ∈ U D . Hence f is U-equivalently continuous. Finally, let Z be the range of the function f and let r > 0 and d ∈ D be given . Since (f × f )(B d i (r i )) because f is a bijection.
Theorem 4.10. An equivalently uniformisable topological space which its associated U-equivalence class generated by a collection of transitive pseudo metrics, can be topologically embedded into a product of transitive pseudo-metric spaces.
Proof: Let (X, T ) be the topological space induced by U D where D is a collection of transitive pseudo metrics on X. By Proposition 4.9, there exists a U-embedding