nLab
closed injections are embeddings

Context

Topology

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

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

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Extra stuff, structure, properties

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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 18:49:56 by Urs Schreiber (92.218.150.85)