Homeomorphism

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces . A function $f:X\to Y$ is a homeomorphism if:

• $f$ is bijective ,
• $f$ is continuous on $X$, and
• $f^{-1}:Y\to X$ is continuous on $Y$.

Homeomorphisms are the isomorphisms in topology: they identify spaces that are “the same up to continuous deformation.” Properties preserved by homeomorphisms are called topological invariants (e.g., compactness , connectedness ).

Examples:

• $f:(0,1)\to\mathbb{R}$ given by $f(x)=\tan(\pi(x-\tfrac12))$ is a homeomorphism (continuous bijection with continuous inverse).
• Any isometry $f:X\to Y$ that is bijective is a homeomorphism.
• The map $f:[0,2\pi)\to S^1\subseteq\mathbb{R}^2$ given by $f(t)=(\cos t,\sin t)$ is continuous and surjective but not injective, hence not a homeomorphism.