TOPOMORPHISM – A NEW APPROACH TO IDENTIFY TOPOLOGIES

Let us start with the topological spaces (X, T) and (Y, T′) where X = {1, 2, 3, ... , 10} , Y = {1, 2, 3, ... , 20} , T = {X, ∅, {1,2,3}} and T = {Y, ∅, {1,2,3,4,5,6}}. These two topological spaces are not homeomorphic as there is no bijection between X and Y . But one can feel that these two spaces are the same in some sense. This feeling motivates us to define the concept of topomorphism. A topomorphism from a topological space (X, T) to a topological space (Y, T′) is defined as a bijection from T to T′ preserving finite intersection and arbitrary union. In this paper we define and discuss the concept of topomorphism. In Section 2 we give all the definitions and notations which we use in this paper, in Section 3 we define topomorphism and investigate certain properties, in Section 4 we study the relationship between topomorphism and homeomorphism, in Section 5 we discuss the concepts of compact subsets and connected subsets of a topological space in the context of topomorphism and in Section 6 we discuss topomorphism in the context of bases for topologies; finally we end the paper with a few concluding remarks motivating further research in this area. DEFINITIONS AND RESULTS

In Section 2 we give all the definitions and notations which we use in this paper, in Section 3 we define topomorphism and investigate certain properties, in Section 4 we study the relationship between topomorphism and homeomorphism, in Section 5 we discuss the concepts of compact subsets and connected subsets of a topological space in the context of topomorphism and in Section 6 we discuss topomorphism in the context of bases for topologies; finally we end the paper with a few concluding remarks motivating further research in this area.

DEFINITIONS AND RESULTS
First we fix the notations.Let ℝ denote the set of real numbers, ∅ denote the empty set.We use the notation   to denote the complement of  in the corresponding space unless there is no ambiguity.
For sets  and , by  −  we denote the set  ∩   .If :  → ′ is a bijection, then for  ∈  ′ , by  −1 () we denote the unique element  ∈  such that () = .For any function :  → , if  ⊆  and  ⊆ , by  → () and  ← () we denote the sets {()/ ∈ } and { ∈ /() ∈  } respectively.These sets are usually denoted by () and  −1 (B); if  and  are a topomorphism and a homeomorphism from (, ) to (, ′), to avoid confusions (for example between (), the image of the element  in  and (), the set of all images of points in ) we use these notations.
A bijective open continuous function from a topological space to a topological space is called a homeomorphism.For terminologies and notations used in this paper which are not mentioned here we refer to [1].However it is worthwhile to state that we assume one point sets are closed in regular spaces and normal spaces.

Topomorphism
We start with the definition of a topomorphism.
(, ) is connected if and only if (,  ′ ) is connected. ii.
(, ) is compact if and only if (,  ′ ) is compact.

Topomorphisms and Homeomorphisms
Let us now see a relation between topomorphism and homeomorphism.Definition 4.1.Let : (, ) → (, ′) be a homeomorphism.Let :  →  ′ be defined as Then  is a topomorphism and is called the topomorphism induced by the homeomorphism .
As a homeomorphism takes open sets to open sets, the function  is well defined; since  is a bijection and and Thus Definition 4.1 is well-defined.
Though every homeomorphism induces a topomorphism, not all topomorphisms are induced by homeomorphisms, even if the two topological spaces have the same cardinality; the topomorphism  of Example 3.2 is not induced by any homeomorphism as  ← ({3}) cannot be an uncountable set under any bijection.Thus the concept of topomorphism is a proper generalization of homeomorphism.
Topomorphism will not preserve the separation properties like Hausdorff and regular as seen in the following example.
But if one point sets are closed in one of the two spaces between which a topomorphism exists, then we get interesting properties.
It is interesting to note that for any  ∈ , () ∉ ( − {}) and () ∈ ( − {}) for all  ∈  other than .The above theorem can be restated as follows: Theorem 4.4.If two spaces are topomorphic and one point sets in one of the spaces are closed, then the other space contains a subspace homeomorphic to the first space.
Furthermore, if : (, ) → (,  ′ ) is a topomorphism and if (, ) is Hausdorff, then, as one point sets in a Hausdorff space are closed, contains a Hausdorff subspace homeomorphic to (, ).The same is true for all topological properties like regular space and normal space.
As every space is a subspace of itself, the assumption "one point sets are closed" is not a necessary condition in the above theorem.However, the importance of the assumption can be seen from the following example.
Then (,   ) and (,   ) are topological spaces.They are topomorphic under the mapping which takes the sets of   to sets of   in the order we listed them above.But there is neither a subspace of  homeomorphic to  nor a subspace of  homeomorphic to .
Let us justify our claim in the example.If possible let (,   ) be a subspace of  homeomorphic to (,   ).As every member of   is of the form  ∩  for some  ∈   , and as cardinality of   and the cardinality of   must be equal,  must contain both 1 and 2; for otherwise the cardinality of   will be less than that of   .Let  = [3,4] ∩ .Then   = {∅, {1}, {1,2}, ,  ∪ {1},  ∪ {1,2}, A}.
There is no homeomorphism between (,   ) and (,   ) because   contains only two finite subsets whereas   contains at least three finite subsets.
There is no homeomorphism between (,   ) and (,   ) because   contains only three finite subsets whereas   contains at least four finite subsets.
Let us consider a converse of Theorem 4.4.If (, ) and (,  ′ ) are spaces in which one point sets are closed and if each of the spaces has a subspace homeomorphic to the other, can we conclude that there is a topomorphism between the spaces?We cannot conclude so.For example each of the topological spaces [0,1] and (0,2), with usual topology, has uncountably many subspaces homeomorphic to the other; but there is no topomorphism between them by Theorem 3.5 as one is compact and the other is not.
We now prove that a bijection from  to  which coincides with a topomorphism is necessarily a homeomorphism.So a bijection between two topological spaces preserving arbitrary union and finite intersection of open sets is necessarily a homeomorphism.
Then  is a homeomorphism from (, ) to (,  ′ ) and the topomorphism induced by  is .
It is easy to see that ( 1) and (2) prove that  −1 and  are continuous as () and  −1 () are open sets in  ′ and .Thus  is a homeomorphism.□

Compactness and Connectedness in the Context of Topomorphisms
Let (, )and (, ′) be topological spaces.Let  and ′ denote the collection of all closed sets of (, ) and (, ′) respectively.With every topomorphism let us associate a function   from  to ′ and study its properties.

3.
is a bijection from   ′ with inverse   −1 defined by Proof.
Other results follow similarly.

i.
If  is an open connected subset of , then () is a connected subset of . ii.
If  is a closed connected subset of , then   () is a connected subset of .
iii.If  is a connected subset of , then   ( ̅ ) is a connected subset of  where  ̅ denotes the closure of   X.
Proof.We prove the third one, as the other results follow similarly.Since  ̅ is closed,   ( ̅ ) is meaningful and is closed in (,  ′ ).As  is connected,  ̅ is also connected.If   ( ̅ ) is not connected, then there exist nonempty disjoint sets  and  , closed in   ( ̅ ) such that   ( ̅ ) =  ∪ .As   ( ̅ ) is closed in ,  and  are closed in  also.So we get which implies that  ̅ is not connected.This is a contradiction to the fact that  ̅ is connected.This completes the proof.……□
If  is an open set in (,  ′ ), then  −1 () is open in (, ) and hence  −1 () =∪   .This implies that  = (∪   ) = ∪ (  ).Thus () is a basis for  ′ .□ Theorem 6.2.Let  and  be nonempty sets, and let  and ′ be bases for topologies  and  ′ on  and  respectively.Let  be a function from  onto ′ having the following properties.
Then  can be extended to a topomorphism from  to ′ uniquely.
This implies that the extension is unique.□ It is easy to see that if the condition ``onto" on  is removed from the statement, then (, ) will be topomorphic to a subspace of (,  ′ ) which is formed by taking the union ∪ () of all sets  ∈ .

CONCLUSION
We have defined and discussed a new concept called topomorphism, as a bijection between topologies which preserve finite intersection and arbitrary union.Topomorphisms in the context of connectedness, compactness, separation axioms and the role of basis in topomorphism were studied deeply.A good theory may be developed by relaxing the condition "bijection" in the definition of topomorphism.Further, a similar theory can be developed in the fuzzy topology theory.Topomorphism do not identify discrete topological spaces on finite sets.A parallel theory may be developed in this context.