( Y, τ Y ) (Y,\tau_Y) a locally compact topological space according to def. ( X, τ X ) (X,\tau_X) be a topological space, ( proper maps to locally compact spaces are closed) Local compactness is one of the conditions that are often required by default for working with topological spaces: locally compact spaces are a class of “ nice topological spaces”. below).Ī locally compact Hausdorff space may also be called a local compactum compare at compactum. There are various definitions in use, which all coincide if the space is also Hausdorff (prop. below) and possibily such that they are topological closures of smaller open neighbourhoods (def. such that one may find them inside every prescribed open neighbourhood (def. Or rather, if one does not at the same time assume that the space is Hausdorff topological space, then one needs to require that these compact neighbourhoods exist in a controlled way, e.g. Homotopy extension property, Hurewicz cofibrationĬlassical model structure on topological spacesĪ topological space is called locally compact if every point has a compact neighbourhood. 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
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