Homeomorphism

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces.

Definition. A homeomorphism is a bijective continuous map $f : X \to Y$ whose inverse $f^{-1} : Y \to X$ is also continuous. The spaces $X$ and $Y$ are homeomorphic if there exists a homeomorphism between them.

Proposition. A bijection $f : X \to Y$ is a homeomorphism if and only if $f$ induces a bijection $\tau_X \to \tau_Y$ given by $U \mapsto f(U)$, i.e., $U$ is open in $X$ if and only if $f(U)$ is open in $Y$.

Proof sketch. If $f$ is a homeomorphism, then $V \in \tau_Y$ iff $f^{-1}(V) \in \tau_X$, so $U \mapsto f(U)$ bijects $\tau_X$ onto $\tau_Y$. Conversely, if $f$ bijects open sets, then both $f$ and $f^{-1}$ preserve open sets, so both are continuous. $\square$

Proposition. Every topological invariant (compactness, connectedness, number of connected components, fundamental group, etc.) is preserved by homeomorphism.

Proof sketch. A property defined purely in terms of the open-set structure is invariant under any bijection that preserves that structure. $\square$

Proposition. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Proof sketch. A closed subset of a compact space is compact; its image under a continuous map is compact; a compact subset of a Hausdorff space is closed. So $f$ maps closed sets to closed sets, hence $f^{-1}$ is continuous. $\square$

Proposition. Homeomorphisms are precisely the isomorphisms in the category $\mathbf{Top}$ of topological spaces and continuous maps.

Examples.

1. $(0, 1) \cong \mathbb{R}$. The map $x \mapsto \tan(\pi x - \pi/2)$ is a homeomorphism.
2. $S^1 \not\cong [0, 1]$. Removing any point from $S^1$ leaves a connected space; removing an interior point from $[0, 1]$ does not. So no homeomorphism exists.
3. Non-example. The map $t \mapsto (\cos 2\pi t, \sin 2\pi t)$ from $[0, 1)$ to $S^1$ is a continuous bijection that fails to be a homeomorphism because removing $0$ from $[0,1)$ disconnects the domain while removing its image does not disconnect $S^1$, so the spaces have different topological invariants.

Remark. In homotopy theory, the weaker notion of homotopy equivalence identifies spaces that are not homeomorphic but have the same “shape” up to continuous deformation. This passage from homeomorphism to homotopy equivalence underlies the move from point-set topology to ∞-categories.