Vol. 34, No. 3, 1970

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Regular-closed spaces and proximities

Douglas Harris

Vol. 34 (1970), No. 3, 675–685

The theory of the compactifications of a completely regular space has been elucidated in recent years by the theory of proximities, introduced by Efremovič and developed especially by Smirnov. The two fundamental results are that a space has a compactification if and only if it has the topology of some proximity, and that there is a one-to-one correspondence from the collection of compactifications of a space onto the collection of proximities that give the topology of the space. We shall generalize these results to a larger class of spaces, which are related to the regular-closed spaces in the same manner as completely regular spaces are related to compact spaces.

Mathematical Subject Classification
Primary: 54.30
Received: 9 June 1969
Revised: 16 January 1970
Published: 1 September 1970
Douglas Harris