Topological manifold with boundary

Definition

A topological manifold with boundary (or simply manifold with boundary) of dimension $n$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 ${\mathbb{R}}^n$ (viz an open set in the space obtained by taking the points in ${\mathbb{R}}^n$ 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 ${\mathbb{R}}^n$. However, the open sets that intersection the boundary will not be open in ${\mathbb{R}}^n$.

Those points on the manifold which do not have a neighbourhood that is homeomorphic to an open set in ${\mathbb{R}}^n$ 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.

Relation with other structures

Stronger structures

• Differential manifold with boundary