nLab
stably compact 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

$(0,1)$ -Category theory
Contents
Idea
Stably compact spaces are topological spaces which share many of the desirable properties of compact Hausdorff spaces , such as compactness and local compactness , without being Hausdorff or even T1 .

They are also a convenient setting for convergence in an ordered setting, being deeply linked to compact ordered spaces .

Definition
A topological space $X$ is called stably compact if the following conditions are met:

Note that the latter notion of coherence is slightly different than the one given at coherent space .

Connection with compact ordered spaces
(…)

Connection with compact regular bitopological spaces
(…) For details see (Jung-Moshier 2006 ).

References
Achim Jung, Stably compact spaces and the probabilistic powerspace construction , ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001 ).

Achim Jung and M. Andrew Moshier. On the bitopological nature of Stone duality. Technical Report CSR-06-13. School of Computer Science, University of Birmingham, December 2006, 110 pages. $[$ pdf $]$

Last revised on May 18, 2022 at 16:48:51.
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