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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
Definition
A topological space is called fully normal if every open cover $\{U_i \subset X\}_{i \in I}$ is a normal cover , i.e., has a refinement by an open cover $\{V_j \subset X\}_{j \in J}$ such that every star (1) in the latter cover is contained in a patch of the former. Furthermore, the resulting cover $\{V_j\}_{j\in J}$ also admits such a star refinement, and this process can be continued indefinitely.

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

(1) $star(x,\mathcal{V})
\;\coloneqq\;
\left\{
V_j \in \mathcal{V} \;\vert\; x \in V_J
\right\}$

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 June 16, 2021 at 08:04:33.
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