nLab
metric spaces are paracompact
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
Statement
Every separable metrisable topological space is paracompact .

With the axiom of choice we have more generally that:

Every metrisable topological space is paracompact .

The orignal proof due to (Stone 48 ) used that metric spaces are fully normal and showed that fully normal spaces are equivalently paracompact (“Stone’s theorem”).

A direct and short proof was later given in (Rudin 68 ).

References
Historically, first it was shown that fully normal spaces are equivalently paracompact in

A. H. Stone, Paracompactness and product spaces , Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (Euclid )
Since it is easy to see that metric spaces are fully normal this implies that metric spaces and hence metriable topological spaces are paracompact. Accordingly this statement came to be known as Stone’s theorem .

A direct and short proof that metric spaces are paracompact was given in

Mary Ellen Rudin? , A new proof that metric spaces are paracompact , AMS 1968 (pdf )
Last revised on May 23, 2017 at 15:12:12.
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