Contents

# Contents

## Statement

###### Proposition

(open/closed continuous injections are embeddings)

A continuous function $f \colon (X, \tau_X) \to (Y,\tau_Y)$ which is

1. an open map or a closed map

###### Proof

If $f$ is injective, then the map onto its image $X \to f(X) \subset Y$ is a bijection. Moreover, it is still continuous with respect to the subspace topology on $f(X)$. Now a bijective continuous function is a homeomorphism precisely if it is an open map or a closed map (by this prop.). But the image projection of $f$ has this property, respectively, if $f$ does (by this prop.).

Created on May 12, 2017 at 18:45:25. See the history of this page for a list of all contributions to it.