nLab
open map

This page is about the concept in topology. For the more general concept see at open 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) between topological spaces 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 is 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.

Example

(projections out of product spaces are open maps)

For (X 1,τ X 1)(X_1,\tau_{X_1}) and (X 2,τ X 2)(X_2,\tau_{X_2}) two topological spaces, then the projection maps

pr i:(X 1×X 2,τ X 1×X 2)(X i,τ X i) pr_i \;\colon\; (X_1 \times X_2, \tau_{X_1 \times X_2}) \longrightarrow (X_i, \tau_{X_i})

out of their product topological space

X 1×X 2 pr 1 X 1 (x 1,x 2) AAA x 1 \array{ X_1 \times X_2 &\overset{pr_1}{\longrightarrow}& X_1 \\ (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_1 }
X 1×X 2 pr 2 X 2 (x 1,x 2) AAA x 2 \array{ X_1 \times X_2 &\overset{pr_2}{\longrightarrow}& X_2 \\ (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_2 }

are open continuous functions (def. 1).

This is because, by definition, every open subset OX 1×X 2O \subset X_1 \times X_2 in the product space topology is a union of products of open subsets U iX 1U_i \in X_1 and V iX 2V_i \in X_2 in the factor spaces

O=iI(U i×V i) O = \underset{i \in I}{\cup} \left( U_i \times V_i \right)

and because taking the image of a function preserves unions of subsets

pr 1(iI(U i×V i)) =iIpr 1(U i×V i) =iIU i. \begin{aligned} pr_1\left( \underset{i \in I}{\cup} \left( U_i \times V_i \right) \right) & = \underset{i \in I}{\cup} pr_1 \left( U_i \times V_i \right) \\ & = \underset{i \in I}{\cup} U_i \end{aligned} \,.

Revised on June 7, 2017 07:34:53 by Urs Schreiber (185.46.137.5)