Abstract
A space satisfies a certain separation axiom only if the topology contains enough open sets to provide disjoint neighborhoods for certain disjoint sets. Compactness, however, limits the number of open sets in a topology, for every open cover of a compact topological space must contain a finite sub-cover. This difference between the separation axioms and the various forms of compactness is illustrated in the extreme by the double pointed finite complement topology (Example 18.7) which is not even T0 yet does satisfy all the forms of compactness.
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© 1978 Springer-Verlag New York Inc.
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Steen, L.A., Seebach, J.A. (1978). Compactness. In: Counterexamples in Topology. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6290-9_3
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DOI: https://doi.org/10.1007/978-1-4612-6290-9_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90312-5
Online ISBN: 978-1-4612-6290-9
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