nLab
coherent topological space
**
topology** (
point-set topology)
see also _
algebraic topology_, _
functional analysis_ and _
homotopy theory_
Introduction
**Basic concepts**
*
open subset,
closed subset,
neighbourhood
*
topological space (see also _
locale_)
*
base for the topology,
neighbourhood base
*
finer/coarser topology
*
closure,
interior,
boundary
*
separation,
sobriety
*
continuous function,
homeomorphism
*
embedding
*
open map,
closed map
*
sequence,
net,
sub-net,
filter
*
convergence
*
category Top
*
convenient category of topological spaces
**[Universal constructions](Top#UniversalConstructions)**
*
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
**
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
*
topological vector space,
Banach space,
Hilbert space
*
topological group
*
topological vector bundle
*
topological manifold
**Examples**
*
empty space,
point space
*
discrete space,
codiscrete space
*
Sierpinski space
*
order topology,
specialization topology,
Scott topology
*
Euclidean space
*
real line,
plane
*
sphere,
ball,
*
circle,
torus,
annulus
*
polytope,
polyhedron
*
projective space (
real,
complex)
*
classifying space
*
configuration space
*
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
*
CW-complexes are paracompact Hausdorff spaces
*
subsets are closed in a closed subspace precisely if they are closed in the ambient space
*
paracompact Hausdorff spaces are normal
*
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
*
Lebesgue number lemma
*
sequentially compact metric spaces are equivalently compact metric spaces
*
compact spaces equivalently have converging subnet of every net
*
sequentially compact metric spaces are totally bounded
*
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
**Theorems**
*
Urysohn's lemma
*
Tietze extension theorem
*
Tychonoff theorem
*
tube lemma
*
Heine-Borel theorem
*
Michael's theorem
*
Brouwer's fixed point theorem
*
topological invariance of dimension
*
Jordan curve theorem
**Basic
homotopy theory**
*
homotopy group
*
covering space
*
Whitehead's theorem
*
Freudenthal suspension theorem
*
nerve theorem
A topological space $X$ is coherent if
This is equivalent to saying that the topos of sheaves $Sh(X)$ on $X$ is a coherent topos.
Revised on May 26, 2010 13:31:54
by
Mike Shulman
(75.3.130.212)