On Bicompact Spaces

Eduard Čech

A chapter in The Mathematical Legacy of Eduard Čech, 1993, pp 38-59 from Springer

Abstract: Abstract The theory of bicompact spaces was extensively studied by P. Alexandroff and P. Urysohn in their paper Mémoire sur les espaces topologiques compacts, Verhandlingen der Kon. Akademia Amsterdam, Deel XIV, No. 1, 1929; I shall refer to this paper with the letters AU. An important result was added by A. Tychonoff in his paper Über die topologische Erweiterung von Räumen, Math. Annalen 102, 1930, who proved that complete regularity is the necessary and sufficient condition for a topological space to be a subset of some bicompact Hausdorff space. As a matter of fact, Tychonoff proves more, viz. that, given a completely regular space S, there exists a bicompact Hausdorff space β(S) such that (i) S is dense in β(S), (ii) any bounded continuous real function defined in the domain S admits of a continuous extension to the domain β(S). It is easily seen that β(S) is uniquely defined by the two properties (i) and (ii). The aim of the present paper is chiefly the study of β(S).

Date: 1993
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DOI: 10.1007/978-3-0348-7524-0_3

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