# Topological manifold with boundary

## Definition

A **topological manifold with boundary** (or simply **manifold with boundary**) of dimension is a topological space satisfying the following:

- It is Hausdorff
- It is second-countable
- Every point has a neighbourhood that is homoemorphic to an open set in the upper half-space (with boundary) of (viz an open set in the space obtained by taking the points in with the last coordinate nonnegative)

Note here that when we say *open set* in the upper half-space we mean open in the relative or subspace topology. Those open sets which do not intersect the boundary of the half-space are also open in . However, the open sets that intersection the boundary will not be open in .

Those points on the manifold which do not have a neighbourhood that is homeomorphic to an open set in *must* coorespond to an open set intersecting the boundary, and moreover, the point itself will lie inside the intersection. Such points are termed boundary points of the manifold.