Abstract
A new concept called topomorphism is defined and discussed in this paper. A topomorphism is defined as a bijection between topologies which preserve finite intersection and arbitrary union. It is proved that topomorphisms preserve connectedness and compactness. Topomorphisms in the context of separation axioms and the role of bases in topomorphism were studied deeply.
Highlights
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}}
It is easy to see that the identity map from T to T is a topomorphism and by Theorem 3.4, the relation ∼ defined on the class of all topological spaces by “(X, T) ∼ (Y, T′) if there is a topomorphism from (X, T) to (Y, T′) ” is an equivalence relation
We prove that a bijection from X to Y which coincides with a topomorphism is necessarily a homeomorphism
Summary
A new concept called topomorphism is defined and discussed in this paper. A topomorphism is defined as a bijection between topologies which preserve finite intersection and arbitrary union. It is proved that topomorphisms preserve connectedness and compactness. Topomorphisms in the context of separation axioms and the role of bases in topomorphism were studied deeply
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