Homeomorphism

Definition

Let $X$ and $Y$ be topological spaces. A function $f \colon X \to Y$ is a homeomorphism when $f$ is bijective, $f$ is continuous, and the inverse function $f^{-1} \colon Y \to X$ is continuous. The spaces $X$ and $Y$ are homeomorphic when there is a homeomorphism $X \to Y$.

Remarks

Equivalently, $f$ is a bijection and, for each $U \subseteq X$, the set $U$ is open if and only if $f(U)$ is open. The map $x \mapsto x / (1 - |x|)$ is a homeomorphism from $(-1, 1)$ to $\mathbb{R}$. A continuous bijection need not be a homeomorphism: $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to the circle is one example. For a compact domain and a Hausdorff target, it is always a homeomorphism. Homeomorphisms keep each topological property.

Explanations

L1 human

A homeomorphism is a way to deform one space into another by stretching and bending, without tearing or gluing. A coffee cup and a doughnut are homeomorphic, but a sphere and a doughnut are not.