Definition:Dimension (Topology)/Locally Euclidean Space

Manifold
Let $$M \ $$ be a manifold.

We say $$M \ $$ 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 $$\phi:U \to \reals^n \ $$.

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