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

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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)