Reference: Introduction to Smooth Manifolds by John M. Lee.
A topological space $M$ is a topological manifold of dimension $n$ if $M$ is locally Euclidean of dimension $n$, where locally Euclidean means that for every point $p \in M$, there exists an open neighborhood $U$ of $p$ in $M$ and a homeomorphism $\varphi: U \to B^n$, then we obtain a set of such charts $\{(U_\alpha, \varphi_\alpha)\}$ called a topological atlas on $M$.
In most cases, we add two more conditions to the definition of a topological manifold:
- $M$ is Hausdorff.
- $M$ is second countable.
$\mathbb R^n, \mathbb P^n$ are topological manifolds. Any open subset or finite product of topological manifolds is a topological manifold.