# Manifold with boundary

From Topospaces

## Definition

A topological space is termed a **manifold with boundary** if it satisfies the following three conditions:

- It is a Hausdorff space.
- It is a second-countable space.
- It is a locally Euclidean half-space: every point has a neighborhood that is homeomorphic to an open subset of Euclidean half-space.

The points that have neighborhoods homeomorphic to open subsets of Euclidean space are termed interior points of the manifold; the other points are termed *boundary points* of the manifold.

The term **manifold with boundary** is sometimes used not for the manifold itself, but for the *pair* comprising the manifold and its boundary (the set of its boundary points).