Topological manifold

Summary

A topological manifold of dimension $n$ is a topological space that looks like the Euclidean space $\R^n$ near each of its points. It is also required to be Hausdorff and second-countable. It is the basic object of geometric topology. Curves, surfaces and the space-times of physics are manifolds, and smooth and piecewise linear manifolds are topological manifolds with extra structure.