# Homeomorphism

A function $f:X \to Y$ is said to be a homeomorphism (topological mapping) if and only if the following conditions are satisfied:

(1) $f$ is bijective
(2) $f$ is continuous
(3) ${f^{ – 1}}$ is continuous

It may be noted that if $f$ is a homeomorphism from $X$ to $Y$, then $X$ is said to be homeomorphic to $Y$ and is denoted by $X \simeq Y$. From the definition of a homeomorphism, it follows that $X$ and $Y$ are homeomorphic spaces, then their points and open sets are put into one-to-one correspondence. In other words, $X$ and $Y$ 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.

Theorems
• Bijective continuous mapping $f:X \to Y$ is open if and only if ${f^{ – 1}}$ is continuous.
• If $X$ and $Y$ are topological spaces, let $X \simeq Y$ mean that $X$ and $Y$ are homeomorphic. Then this relation is reflexive, symmetric and transitive.
• Let $X$ and $Y$ be topological spaces and $f:X \to Y$ be a bijective function, then the following are equivalent: (1) $f$ is a homeomorphism; (2) for any subset $U$ of $X$, $f\left( U \right)$ is open in $Y$ if and only if $U$ is open in $X$; (3) for any subset $C$ of $X$, $f\left( C \right)$ is closed in $Y$ if and only if $C$ is closed in $X$; (4) for any subset $A$ of $X$, $f\left( {\overline A } \right) = \overline {f\left( A \right)}$.