nLab
quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff

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

(quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff)

Let

π:(X,τ X)(Y,τ Y) \pi \;\colon\; (X, \tau_X) \longrightarrow (Y, \tau_Y)

be a continuous function between topological spaces such that

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

  2. π\pi is a surjection and τ Y\tau_Y is the corresponding quotient topology.

Then the following are equivalent

  1. (Y,τ Y)(Y, \tau_Y) is itself a Hausdorff topological space;

  2. π\pi is a closed map.

Proof

The implicaton ((Y,τ Y)Hausdorff)(πclosed)\left( (Y, \tau_Y)\, \text{Hausdorff} \right) \Rightarrow \left( \pi \, \text{closed} \right) follows since maps from compact spaces to Hausdorff spaces are closed. We need to show the converse.

Hence assume that π\pi is a closed map. We need to show that for every pair of distinct points y 1y 2Yy_1 \neq y_2 \in Y there exist open neighbourhoods U y 1,U y 2τ YU_{y_1}, U_{y_2} \in \tau_Y which are disjoint, U y 1U y 2=U_{y_1} \cap U_{y_2} = \emptyset.

First notice that the singleton subsets {x},{y}Y\{x\}, \{y\} \in Y are closed. This is because they are images of singleton subsets in XX, by surjectivity of ff, and because singletons in a Hausdorff space are closed, and because images under ff of closed subsets are closed, by the assumption that ff is a closed map.

It follows that the pre-images

C 1π 1({y 1})AAC 2π 1({y 2}). C_1 \coloneqq \pi^{-1}(\{y_1\}) \phantom{AA} C_2 \coloneqq \pi^{-1}(\{y_2\}) \,.

are closed subsets of XX.

Now since compact Hausdorff spaces are normal it follows that we may find disjoint open subset U 1,U 2τ XU_1, U_2 \in \tau_X such that

C 1U 1AAAC 2U 2. C_1 \subset U_1 \phantom{AAA} C_2 \subset U_2 \,.

Moreover, by this lemma we may find these U iU_i such that they are both saturated subsets. Therefore finally this lemma says that the images π(U i)\pi(U_i) are open in (Y,τ Y)(Y,\tau_Y). These are now clearly disjoint open neighbourhoods of y 1y_1 and y 2y_2.

Applications

Example

Consider the function

[0,2π]/ S 1 2 t (cos(t),sin(t)) \array{ [0,2\pi]/\sim &\longrightarrow& S^1 \subset \mathbb{R}^2 \\ t &\mapsto& (cos(t), sin(t)) }

This is clearly a continuous function and a bijection on the underlying sets. Moreover, since continuous images of compact spaces are compact and since the closed interval [0,1][0,1] is compact we also obtain another proof that the circle is compact.

Hence since continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms the above map is in fact a homeomorphism

[0,2π]/S 1. [0,2\pi]/\sim \;\simeq\; S^1 \,.
Revised on May 28, 2017 14:08:48 by Urs Schreiber (92.218.150.85)