Contents

# Contents

## Idea

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

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

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

For details see manifold.

## Properties

###### Proposition

(manifolds with boundary form full subcategory of diffeological spaces)

The evident functor

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

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

## References

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

Related discussion to the case of manifolds with corners is in

Last revised on August 3, 2018 at 12:24:19. See the history of this page for a list of all contributions to it.