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Definition: Let $X$ and $Y$ be topological spaces. A continuous map $f:X\to Y$ is a homeomorphism if there exists a continuous map $g:Y\to X$ so that $g\circ f=\mbox{id}_{X}$ and $g\circ f=\mbox{id}_{Y}.$


Definition: An $n$ dimensional topological manifold is a topological space $M$ such that for each $x\in M,$ there exists an open neighborhood $U$ of $x$ so that $U$ is homeomorphic to an open subset of $\mathbb{R}^{n}.$

註:如果你不太清楚什麼是topological space(拓樸空間)，你可以使用metric space (賦距空間)來替代。