nLab Heine-Cantor theorem

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

Analysis Theorems

Idea

The Heine-Cantor theorem states that, given a pointwise continuous function $f \,\colon\, M \to N$ between metric spaces, that if $M$ is compact then $f$ is uniformly continuous.

Last revised on June 24, 2024 at 11:00:15. See the history of this page for a list of all contributions to it.