nLab closed injections are embeddings

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Statement

Proposition

(open/closed continuous injections are embeddings)

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

  1. an injective function

  2. an open map or a closed map

is a topological embedding.

Proof

If ff is injective, then the map onto its image Xf(X)YX \to f(X) \subset Y is a bijection. Moreover, it is still continuous with respect to the subspace topology on f(X)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 ff has this property, respectively, if ff does (by this prop.).

Created on May 12, 2017 at 22:45:25. See the history of this page for a list of all contributions to it.