# homeomorphisms preserve connected components

Let $X,Y$ be topological spaces and $X=\bigcup\,X_{i}$, $Y=\bigcup\,Y_{j}$ be decompositions into connected components.

Assume that $f:X\to Y$ is a homeomorphism. Then for any $i$ there exists $j$ such that $f(X_{i})=Y_{j}$.

Proof. Take any $i$. Because $f$ is continuous $f(X_{i})$ is connected, then there exists $j$ such that $f(X_{i})\subseteq Y_{j}$ (because $Y_{j}$ is a connected component). Now $f$ is a homeomorphism, $f^{-1}(Y_{j})\cap X_{i}\neq\emptyset$, $Y_{j}$ is connected and $X_{i}$ is a connected component, so $f^{-1}(Y_{j})\subseteq X_{i}$. Thus $Y_{j}\subseteq f(X_{i})$, which completes the proof. $\square$

Title homeomorphisms preserve connected components HomeomorphismsPreserveConnectedComponents 2013-03-22 18:45:32 2013-03-22 18:45:32 joking (16130) joking (16130) 5 joking (16130) Derivation msc 54D05