nLab
locally finite cover

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Contents

Idea

A locally finite cover is a cover which in a suitable sense looks locally like a finite cover.

Definition

Definition

(locally finite cover)

Let (X,τ)(X,\tau) be a topological space.

A cover {U iX} iI\{U_i \subset X\}_{i \in I} of XX by subsets of XX is called locally finite if it is a locally finite set of subsets, hence if for all points xXx \in X, there exists a neighbourhood U x{x}U_x \supset \{x\} such that it intersects only finitely many elements of the cover, hence such that U xU iU_x \cap U_i \neq \emptyset for only a finite number of iIi \in I.

If {U iX} iI\{U_i \subset X\}_{i \in I} is an open cover, then it is called a locally finite open cover.

Remark

(alternative characterizations of local finiteness)

Let XX be a topological space and let {U iX} iI\{U_i \to X\}_{i \in I} be a cover by subsets. Then the following are equivalent:

  1. {U iX} iI\{U_i \subset X\}_{i \in I} is locally finite (def. 1);

  2. there exist an open cover {V jX} jJ\{V_j \subset X\}_{j \in J} such that for each jJj \in J there is a finite number of iIi \in I that V jV_j intersects V iV_i.

This is because the various V iV_i constitute open neighbourhoods for all points xXx \in X.

Moreover, suppose that {V jX} jJ\{V_j \subset X\}_{j \in J} is a cover by any subsets (not necessarily open), but that it is itself a locally finite set of subsets. Then if for all jJj \in J there are a finite number of iIi \in I such that U iU_i intersects V jV_j, it follows again that also {U iX} iI\{U_i \subset X\}_{i \in I} is locally finite.

This is because by the local finiteness of {V jX} jJ\{V_j \subset X\}_{j \in J} we have for every point xXx \in X an open neighbourhood O x{x}O_x \supset \{x\} which intersects only a finite number of the V jV_j, and since each of these intersects only a finite number of the U iU_i, in total also O xO_x can only intesect a finite number of the U iU_i.

Properties

The following says that if there exists a locally finite refinement of a cover, then in fact there exists one with the same index set as the original cover.

Lemma

(locally finite refinement induces locally finite cover with original index set)

Let (X,τ)(X,\tau) be a topological space, let {U iX} iI\{U_i \subset X\}_{i \in I} be an open cover, and let {V jX} jJ\{V_j \subset X\}_{j \in J}, be a refinement to a locally finite cover.

By definition of refinement we may choose a function

ϕ:JI \phi \colon J \to I

such that

jJ(V jU ϕ(j)). \underset{j \in J}{\forall}\left( V_j \subset U_{\phi(j)} \right) \,.

Then {W iX} iI\left\{ W_i \subset X \right\}_{i \in I} with

W i{jϕ 1({i})V j} W_i \;\coloneqq\; \left\{ \underset{j \in \phi^{-1}(\{i\})}{\cup} V_j \right\}

is still a refinement of {U iX} iI\{U_i \subset X\}_{i \in I} to a locally finite cover.

Proof

It is clear by construction that W iU iW_i \subset U_i, hence that we have a refinement. We need to show local finiteness.

Hence consider xXx \in X. By the assumption that {V jX} jJ\{V_j \subset X\}_{j \in J} is locally finite, it follows that there exists an open neighbourhood U x{x}U_x \supset \{x\} and a finite subset KJK \subset J such that

jJ\K(U xV j=). \underset{j \in J\backslash K}{\forall} \left( U_x \cap V_j = \emptyset \right) \,.

Hence by construction

iI\ϕ(K)(U xW i=). \underset{i \in I\backslash \phi(K)}{\forall} \left( U_x \cap W_i = \emptyset \right) \,.

Since the image ϕ(K)I\phi(K) \subset I is still a finite set, this shows that {W iX} iI\{W_i \subset X\}_{i \in I} is locally finite.

Lemma

(shrinking lemma)

Let (X,τ)(X,\tau) be a normal topological space, and let {U iX} iI\{U_i \subset X\}_{i \in I} be a locally finite open cover. Then there exists a shrinking to a locally finite open cover {V iX} iI\{V_i \subset X\}_{i \in I} whose closures Cl()Cl(-) are still contained in the original cover:

V iCl(V i)U i. V_i \subset Cl(V_i) \subset U_i \,.

Revised on May 25, 2017 13:48:56 by Urs Schreiber (92.218.150.85)