A function is said to be a homeomorphism (topological mapping) if and only if the following conditions are satisfied:
(1) is bijective
(2) is continuous
(3) is continuous
It may be noted that if is a homeomorphism from to , then is said to be homeomorphic to and is denoted by . From the definition of a homeomorphism, it follows that and are homeomorphic spaces, then their points and open sets are put into one-to-one correspondence. In other words, and differ only in the nature of their points, but from the point of view of the subject of topology they are identical or have the same topological structure.
Remarks: “Homeomorphism” helps reduce complicated problems into simple form, that is, an apparently complicated space may possibly be homeomorphic to some space more familiar to us. Hence in this way, one determines the properties of complicated spaces easily.
• Bijective continuous mapping is open if and only if is continuous.
• If and are topological spaces, let mean that and are homeomorphic. Then this relation is reflexive, symmetric and transitive.
• Let and be topological spaces and be a bijective function, then the following are equivalent: (1) is a homeomorphism; (2) for any subset of , is open in if and only if is open in ; (3) for any subset of , is closed in if and only if is closed in ; (4) for any subset of , .