Topological manifold

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 .

An $n$ -dimensional topological manifold is a Hausdorff, second-countable topological space $M$ such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb R^n$ .

Charts and smooth structures

The local homeomorphisms are coordinate charts; their collection covers $M$ .

The integer $n$ is the dimension of $M$ . No differentiability is assumed. A smooth manifold is a topological manifold equipped with charts whose transition maps are smooth, so every smooth manifold has an underlying topological manifold but not conversely.