REVERSIBLE ČECH CLOSURE SPACES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper we present the notion of reversibility in Čech-closure spaces. Some characterization theorems are obtained similar to topological spaces. The relation between reversible Čech-closure spaces with the underlying topological spaces, complete homogeneity and reversibility in Čech closure spaces are also investigated.


INTRODUCTION
The concept ofČech closure operators on a set T,was introduced by EdwardČech [2], by genaralizing the notion of Kuratowski closure operators (topological closure operators). AČech closure space (T, µ) is a set T with aČech closure operator µ, which need not satisfy the idempotent law of topological closure. Many properties which hold in topological spaces hold inČech closure spaces as well. Using the above information, in this paper we extend the notion of reversibility, introduced by M. Rajagopalan and A. Willansky [7] and check how far the results good inČech closure spaces.

PRELIMINARIES
Let T be a set, and P(T) denotes the power set of T, a mapping µ : P(T) −→ P(T) is called aČech closure operator provided it satisfies the following three axioms .
(2) µ(φ ) = φ Then T,together with the aČech closure operator µ is calledČech closure space and is denoted by (T, µ). We denote aČech closure space throughout this paper as closure space for convenience. If µ(µ(A )) = µ(A ), ∀A ⊂ T, then (T, µ) is a topological space. Thus the concept of closure spaces, generalizes the definition of topological spaces.
The closed sets in a closure space (T, µ) are those subset F of T, such that F = µ(F ) and a subset O of T is open provided its complement T − O is closed.
Note that the collection of all open sets in a closure space is a topology on T, denoted by τ(µ), and τ(µ) is the topology associated with the closure space (T, µ).
In a closure space (T, µ), µ is finitely generated, provided for any subset A of T, µ(A ) = {µ(a); a ∈ A }, in this scenario (T, µ) is called finitely generated closure space. Also every finitely generated closure operator is characterised by its action on singleton sets.
In [9] it is proved that the converse of the above rseult not always true.
A closure µ is said to be coarser (weaker) than a closure ν on the same set for each subset Y of T, we denote it by µ < ν. If µ is coarser than ν we also say ν is finer (stronger) than µ.
Let (T, µ) and (U, ν) be closure spaces. A map θ : (T, µ) −→ (U, ν) is said to be closed (resp. open) if θ (F ) is a closed ( resp. open ) subset of (U, ν) whenever F is a closed (resp. open) subset of (T, µ). Every one to one , ontoČ-continuous closed (open) map from a closure space to any other closure space is aČ-homeomorphism. Theorem 2.3. A necessary and sufficient condition for a closure space (T, µ) to be finer than the closure space (T, ν) is that the identity map i : (T, µ) −→ (T, ν) isČ-continuous, [1].
The subspace of a closure space is defined as follows. If (T, µ) be a closure space and W ⊂ T, then a closure µ on W is defined as µ (Y ) = W ∩ µ(Y ) for all Y ⊆ W. Then the closure space (W, µ ) is called the subspace of (T, µ).

REVERSIBLE CLOSURE SPACES
In [7], Rajagopalan and Willansky unified the notion of minimal and maximal topologies by introducing reversible topological spaces. Also in [7], they proved that a topological space (P, τ) is reversible, if the only continuous self-bijections on P are the homeomorphisms. As an analogous way we define reversibleČech -closure space as follows. Proof. Let (T, µ 1 ), (T, µ 2 ), (U, ν), (U, ν ) are closure spaces with µ 1 and ν finer than µ 2 and ν .
To prove the second statement, assume ρ be aČ-continuous map from (T, µ 2 ) to (U, ν). Also The following theorem characterises a reversible closure space. Analogous result and proof in topological context are found in [7]. Theorem 3.3. The necessary and sufficient condition for a closure space (T, µ) to be reversible is that, each one to one, ontoČ-continuous map of the space onto itself is aČ-homeomorphism.
Proof. Assume (T, µ) is a reversible closure space and θ : (T, µ) −→ (T, µ) is aČ-continuous function, which is both one to one and onto. Let ν be a closure operator on T such that F ⊆ T is closed in (T, ν), if µ(θ (F )) = θ (F ). Then the closure ν on T weaker than µ, since, let K be any closed set in (T, ν) then θ (K ) is closed in (T, µ) again since, θ is continuous Conversely assume each self bijectiveČ-continuous map on (T, µ) is aČ-homeomorphism.
To prove this, we first prove (N, µ)  In this example, the associated topological space is the indiscrete space and hence the associated topological space is also reversible.
Proposition 3.6. If T is finite and µ is any closure operator on T, then (T, µ) is a reversible closure space.
Proof. Consider the set of all bijectiveČ-continuous functions on T, we denote this set by G, and let θ ∈ G. Now using Theorem 2.1, we have θ 2 ∈ G, similarly θ 3 , θ 4 ... ∈ G. Since T is finite, there exist n > m ∈ Z + such that θ n = θ m in G. Hence θ n−m = id T , it follows that θ n−m−1 = θ −1 ∈ G. Thus θ is aČ-homeomorphism. Hence by Theorem 3.3, T is reversible.
Given below are some examples of non-reversible closure spaces. EXAMPLE 3.11. A non reversible topological space can be viewed as a non reversible closure space. EXAMPLE 3.12. Define a closure operator µ on Z, as follows ∀k ∈ Z, µ(k) = k if k < 0 and Then (Z, µ) is a closure space (not topological), which is not reversible.  Remark 3.14. From the above example we have the following proposition.
Then (Z, µ m,n ) is a closure space, which is not reversible.
For, take m = 1, n = 2 and consider the function θ on Z as θ (n) = n + 1, then θ is one to one, onto andČ-continuous but not aČ-homeomorphism. Hence from Theorem 3.3, (Z, µ m,n ) is not reversible.
Remark 3.17. The above example gives a non reversible non topological closure space, whose associated topological space ( in this example the topology is the indiscrete topology) is reversible. Also from the above example we have the following proposition.
Definition 3.18. [8] Let X be a set, for x, y ∈ X x = y,define µ x,y on P(X) as follows Then µ x,y is a closure opertaor on X, called the infra closure operator on X.
Proposition 3.19. The infra closure operator on Z is non reversible.
Proof. Follows from Example 3.16.
It is known that, in topological spaces, reversibility is a topological property [7]. We state an equivalent result in closure spaces context as follows.
In [4] Larson studied the concept of complete homogeneity in topological spaces and characterized all spaces which are minimum and maximum with respect to a topological property. He also determined a characterization theorem for completely homogeneous topological spaces. In an analogous way Ramachandran [8], studied complete homogeneity in closure spaces.  of T on to itself is aČ-homeomorphism [8].
The following theorem finds the relation between complete homogeneity and reversibility in closure spaces.
Some result which are not generally valid in topological spaces are the same in closure spaces even, in view of Proposition 3.8. In [7], it is showed that reversibility is not a hereditary in topological spaces, whereas complete homogeneity is hereditary in topological spaces [4]. Using these facts, we get the proposition below. In topology, spaces that are reversible, which are maximal or minimal in relation to some defined topological property. A compact Hausdorff space, for example,is maximal compact and minimal Hausdorff. Such space is just one example of a reversible space. In this section we try an analogous study in reversible closure spaces. (1) (T, µ) is reversible.
Hence µ must be minimum for some P.
To prove (1) and (3) are equivalent a similar proof will work, if we redefine the property P as, a closure space (U, ν) has a property P, if there exist a one to one , onto,Č-continuous map from (T, µ) to (U, ν).
Analogous theorem and proof in complete homogeneity context(topological) are found in [4].

CONCLUDING REMARKS
In this paper we define reversible closure space analogous to reversible topological spaces.
We obtain some results analogous to topological spaces. The relation between reversible topological spaces and the underlying topological spaces is obtained. We prove not every finitely generated closure space is reversible also find a relation between complete homogeneity and reversibility in closure space context. We also find some non topological non reversible closure spaces whose associated topological space is reversible. These are some questions for further investigation.
(1) Identify the lattice of reversible closure operator and investigate its properties.
(2) Does there exist a reversible non topological closure space which is the union of two disjoint non reversible closure spaces ? The answer to this question in topological context is yes.
(3) Identify and study reversible non topological closure operators in detail.
(4) Extend this study in fuzzy context is also recommended.