Definition:Dimension (Topology)/Locally Euclidean Space

Manifold
Let $M$ be a manifold.

$M$ has dimension $n$ if, for every point $x \in M, \exists U \in \vartheta_M$ such that $\vartheta_M$ is the topology of $M$, $x \in U$, and there exists a homeomorphism $\phi:U \to \R^n$.

In the context of smooth manifolds and differential topology, the homeomorphism above is strengthened to a diffeomorphism.