Homotopy equivalence, deformation retract Second-countable regular spaces are paracompactĬW-complexes are paracompact Hausdorff spaces Locally compact and second-countable spaces are sigma-compact Locally compact and sigma-compact spaces are paracompact Injective proper maps to locally compact spaces are equivalently the closed embeddings Proper maps to locally compact spaces are closed Paracompact Hausdorff spaces equivalently admit subordinate partitions of unity Sequentially compact metric spaces are totally boundedĬontinuous metric space valued function on compact metric space is uniformly continuous Sequentially compact metric spaces are equivalently compact metric spaces Quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is HausdorffĬompact spaces equivalently have converging subnet of every net Open subspaces of compact Hausdorff spaces are locally compact Line with two origins, long line, Sorgenfrey lineĬontinuous images of compact spaces are compactĬlosed subspaces of compact Hausdorff spaces are equivalently compact subspaces Mapping spaces: compact-open topology, topology of uniform convergence Order topology, specialization topology, Scott topology Topological vector bundle, topological K-theory Topological vector space, Banach space, Hilbert space Simply-connected space, locally simply-connected space Second-countable space, first-countable spaceĬontractible space, locally contractible space Sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact Kolmogorov space, Hausdorff space, regular space, normal space Metric space, metric topology, metrisable space convenient category of topological spaces.Open subset, closed subset, neighbourhoodīase for the topology, neighbourhood base

See also differential topology, algebraic topology, functional analysis and topological homotopy theory Topology ( point-set topology, point-free topology)

Subspace definition definition