Homeomorphism - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A homeomorphism between topological spaces $X$ and $Y$ is a bijective function $f:X\to Y$ such that $f$ is a continuous map and its inverse function $f^{-1}:Y\to X$ is also continuous.

Remarks

If there exists a homeomorphism between $X$ and $Y$ , the spaces are called homeomorphic; this means they are “the same” from the topological viewpoint, sharing properties like compactness and connectedness.

Examples

The map $f:(0,1)\to\mathbb{R}$ given by $f(x)=\tan(\pi(x-\tfrac12))$ is a homeomorphism.
The map $g:\mathbb{R}\to(0,\infty)$ given by $g(x)=e^x$ is a homeomorphism.
If two spaces have exactly the same open sets, the identity map between them is a homeomorphism.