nLab
fully normal topological space

Contents

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

A topological space is called fully normal if every open cover {U iX} iI\{U_i \subset X\}_{i \in I} is a normal cover, i.e., has a refinement by an open cover {V jX} jJ\{V_j \subset X\}_{j \in J} such that every star in the latter cover is contained in a patch of the former. Furthermore, the resulting cover {V j} jJ\{V_j\}_{j\in J} also admits such a star refinement, and this process can be continued indefinitely.

Here, for xXx \in X a point, the star of xx is the union of the patches that contain xx:

star(x,𝒱){V j𝒱|xV J} star(x,\mathcal{V}) \;\coloneqq\; \left\{ V_j \in \mathcal{V} \;\vert\; x \in V_J \right\}

Normal covers are also known as numerable covers, since they are precisely the open covers that admit a subordinate partition of unity.

In pointfree topology

Any completely regular locale has a largest uniformity, the fine uniformity, which consists of all normal covers.

If a completely regular locale admits a complete uniformity?, then the fine uniformity is complete.

A locale is paracompact if and only if it admits a complete uniformity?. In this case, we can take the fine uniformity.

Examples

Properties

Last revised on May 20, 2020 at 06:27:46. See the history of this page for a list of all contributions to it.