Let \(X\) and \(Y\) be topological spaces. A bijection \(f\colon X\to Y\) is a homeomorphism if both \(f\) and \(\inverseof{f}\colon Y\to X\) are continuous. The spaces \(X\) and \(Y\) are homeomorphic, written \(X\cong Y\), if there exists a homeomorphism \(X\to Y\).
For a bijection \(f\colon X\to Y\) the following are equivalent: \(f\) is a homeomorphism; \(f\) is continuous and open; for every \(U\subseteq X\), \(U\) is open in \(X\) if and only if \(f(U)\) is open in \(Y\). A homeomorphism therefore induces a bijection between the topologies of \(X\) and \(Y\), and every property defined in terms of open sets holds for \(X\) if and only if it holds for \(Y\). An injective continuous map \(f\colon X\to Y\) that is a homeomorphism onto \(f(X)\) with the subspace topology is a topological embedding.
[@Mun00].