Vol. 82, No. 1, 1979

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PM-normality and the insertion of a continuous function

Ernest Paul Lane

Vol. 82 (1979), No. 1, 155–162

Spaces in which each regular closed subset is an intersection of a sequence of closed neighborhoods are investigated. This property is shown to be equivalent to each of the following: Each regular closed subset is a zero set of a continuous function. Each normal lower semicontinuous function defined on the space is a limit of an increasing sequence of continuous functions. The space satisfies the strong C insertion property for normal semicontinuous functions. Separation properties of X × I, which are weaker than normality, are related to the insertion of a continuous function between two comparable functions defined on X.

Mathematical Subject Classification 2000
Primary: 54D15
Secondary: 54C50
Received: 23 August 1978
Revised: 20 November 1978
Published: 1 May 1979
Ernest Paul Lane