nLab compact metrizable space

Context

Topology

topology (point-set topology, point-free topology)

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
- convenient category of topological spaces

Universal constructions

initial topology, final topology
subspace, quotient space,
fiber space, space attachment
product space, disjoint union space
mapping cylinder, mapping cocylinder
mapping cone, mapping cocone
mapping telescope
colimits of normal spaces

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
- real line, plane
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
- loop space, path space
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

Hausdorff spaces are sober
schemes are sober
continuous images of compact spaces are compact
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces are normal
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
closed injections are embeddings
proper maps to locally compact spaces are closed
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and sigma-compact spaces are paracompact
locally compact and second-countable spaces are sigma-compact
second-countable regular spaces are paracompact
CW-complexes are paracompact Hausdorff spaces

Theorems

Urysohn's lemma
Tietze extension theorem
Tychonoff theorem
tube lemma
Michael's theorem
Brouwer's fixed point theorem
topological invariance of dimension
Jordan curve theorem

Analysis Theorems

Heine-Borel theorem
intermediate value theorem
extreme value theorem

topological homotopy theory

left homotopy, right homotopy
homotopy equivalence, deformation retract
fundamental group, covering space
fundamental theorem of covering spaces
homotopy group
weak homotopy equivalence
Whitehead's theorem
Freudenthal suspension theorem
nerve theorem
homotopy extension property, Hurewicz cofibration
cofiber sequence
Strøm model category
classical model structure on topological spaces

Analysis

analysis (differential/integral calculus, functional analysis, topology)

epsilontic analysis

infinitesimal analysis

computable analysis

Introduction

Basic concepts

triangle inequality

metric space, normed vector space

open ball, open subset, neighbourhood

metric topology

sequence, net

convergence, limit of a sequence

compactness, sequential compactness

differentiation, integration

topological vector space

Basic facts

continuous metric space valued function on compact metric space is uniformly continuous

…

Theorems

intermediate value theorem

extreme value theorem

Heine-Borel theorem

…

Definition
Related concepts
References

Definition

A compact metrizable space is a metrizable topological space that is also a compact space.

Remark

(Terminology)
Compact metrizable spaces are also called compacta (singular: compactum).

But be warned that this is ambiguous, as “compacta” is also used to refer to compact Hausdorff spaces. This latter usage is not as wide-spread but has been adopted in many nLab articles.

Related concepts

metrizable topological space
compact topological space

References

References using the term “compactum” for compact metric spaces:

Encyclopedia of Mathematics: Compactum
WolframMathWorld: Compactum
Paul Bankston: Base-free Formulas in the Lattice-theoretic Study of Compacta, Arch. Math. Logic 50 (2011) 531–542 [doi:10.1007/s00153-011-0230-2, epub:1145]

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