nLab
coherent topological space

**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](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 * 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

A topological space XX is coherent if

This is equivalent to saying that the topos of sheaves Sh(X)Sh(X) on XX is a coherent topos.

Revised on May 26, 2010 13:31:54 by Mike Shulman (75.3.130.212)