closed injections are embeddings (changes) in nLab

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

  • 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

  • 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

• 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