nLab
closed map

This page is about the concept in topology. For the more general concept see at closed morphism.

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

Definition

Definition

(open maps and closed maps)

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

Examples

Example

(image projections of open/closed maps are themselves open/closed)

If a continuous function f:(X,τ X)(Y,τ Y)f \colon (X,\tau_X) \to (Y,\tau_Y) is an open map or closed map (def. 1) then so its its image projection Xf(X)YX \to f(X) \subset Y, respectively, for f(X)Yf(X) \subset Y regarded with its subspace topology.

Proof

If ff is an open map, and OXO \subset X is an open subset, so that f(O)Yf(O) \subset Y is also open in YY, then, since f(O)=f(O)f(X)f(O) = f(O) \cap f(X), it is also still open in the subspace topology, hence Xf(X)X \to f(X) is an open map.

If ff is a closed map, and CXC \subset X is a closed subset so that also f(C)Yf(C) \subset Y is a closed subset, then the complement Y\f(C)Y \backslash f(C) is open in YY and hence (Y\f(C))f(X)=f(X)\f(C)(Y \backslash f(C)) \cap f(X) = f(X) \backslash f(C) is open in the subspace topology, which means that f(C)f(C) is closed in the subspace topology.

Proposition

(maps from compact spaces to Hausdorff spaces are closed and proper)

Let f:(X,τ X)(Y,τ Y)f \colon (X, \tau_X) \longrightarrow (Y, \tau_Y) be a continuous function between topological spaces such that

  1. (X,τ X)(X,\tau_X) is a compact topological space;

  2. (Y,τ Y)(Y,\tau_Y) is a Hausdorff topological space.

Then ff is

  1. a closed map (def. 1);

  2. a proper map.

Proposition

(proper maps to locally compact spaces are closed)

Let

  1. (X,τ X)(X,\tau_X) be a topological space,

  2. (Y,τ Y)(Y,\tau_Y) a locally compact topological space according to def. \ref{LocallyCompactSpace},

  3. f:XYf \colon X \to Y a continuous function.

Then:

If ff is a proper map, then it is a closed map.

Revised on May 12, 2017 17:38:31 by Urs Schreiber (46.183.103.8)