Homotopy extension property, Hurewicz cofibrationĬlassical model structure on topological spaces 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)
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Part II: Introduction to Topology 2 – Basic Homotopy Theory \ \ \, ( pdf 61p)įor introduction to abstract homotopy theory see instead at Introduction to Homotopy Theory. Part I: Introduction to Topology 1 – Point-set Topology \ \ \ ( pdf 203p)
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\ \ \ \ \ \ \ (web version requires Firefox browser – free download) The second part introduces some basics of homotopy theory, mostly the fundamental group, and ends with their first application to the classification of covering spaces.
We close with discussion of the basics of topological manifolds and differentiable manifolds, laying the foundations for differential geometry. In passing, some basics of category theory make an informal appearance, used to transparently summarize some conceptually important aspects of the theory, such as initial and final topologies and the reflection into Hausdorff and sober topological spaces. Starting from scratch (required background is just a basic concept of sets), and amplifying motivation from analysis, it first develops standard point-set topology ( topological spaces). This page contains a detailed introduction to basic topology.
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