nLab
manifold with boundary

Contents

Idea

A manifold is a topological space that is locally isomorphic to a Cartesian space n\mathbb{R}^n.

A manifold with boundary is a topological space that is locally isomorphic either to an n\mathbb{R}^n or to a half-space H n={x n|x n0}H^n = \{ \vec x \in \mathbb{R}^n | x^n \geq 0\}.

A manifold with corners is a topological space that is locally isomorphic to an H i n={x n|x i,x i+1,,x n0}H^n_i = \{ \vec x \in \mathbb{R}^n | x^i , x^{i+1}, \cdots, x^n \geq 0\} for 0in0 \leq i \leq n.

For details see manifold.

Properties

Proposition

(manifolds with boundary form full subcategory of diffeological spaces)

The evident functor

SmthMfdWBdrAAAADiffeologicalSpaces SmthMfdWBdr \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces

from the category of smooth manifolds with boundary to that of diffeological spaces is fully faithful, hence is a full subcategory-embedding.

(Igresias-Zemmour 13, 4.16)

References

The full subcategory-embedding of manifolds with boundary into that of diffeological spaces is discussed in

Last revised on June 18, 2018 at 08:01:42. See the history of this page for a list of all contributions to it.