Compactness

Lynn Arthur Steen and J. Arthur Seebach
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Lynn Arthur Steen: Saint Olaf College
J. Arthur Seebach: Saint Olaf College

Chapter Section 3 in Counterexamples in Topology, 1978, pp 18-27 from Springer

Abstract: 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.

Date: 1978
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DOI: 10.1007/978-1-4612-6290-9_3

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