nLab
locally finite cover

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Contents

Idea

A cover which is locally a finite cover.

Definition

Definition

(locally finite cover)

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

An open cover {U iX} iI\{U_i \subset X\}_{i \in I} of XX is called locally finite 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.

Properties

Any open cover defined by a (generalized) partition of unity has a locally finite shrinking. (…details…)

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 (ϕ:JI,{V jX} jJ)(\phi \colon J \to I, \{V_j \subset X\}_{j \in J}), be a refinement to a locally finite cover.

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 finitee 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

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

Revised on April 28, 2017 08:19:43 by Urs Schreiber (92.218.150.85)