# Topological manifold with boundary

(Redirected from 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 $\R^n$ (viz an open set in the space obtained by taking the points in $\R^n$ 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 $\R^n$. However, the open sets that intersection the boundary will not be open in $\R^n$.

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