Manifold with boundary

A smooth manifold with boundary has charts in the half-space $\{x_n\ge 0\}\subseteq {\mathbb{R}}^n$, with smoothly compatible transition maps. The boundary is the locus modeled on $x_n=0$.

A collar neighborhood identifies a neighborhood of the boundary with $\partial W\times [0,\epsilon )$, with the boundary at parameter zero. It allows a manifold's boundary to be pushed slightly inward by a homotopy.