nLab
locally finite cover

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

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 U iU_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 June 24, 2017 01:29:52 by Anonymous (2604:2000:e100:9200:54ec:c8c0:d6f1:6df9)