Vol. 59, No. 2, 1975

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𝜃-closed subsets of Hausdorff spaces

Raymond Frank Dickman and Jack Ray Porter

Vol. 59 (1975), No. 2, 407–415

A topological property of subspaces of a Hausdorff space, called 𝜃-closed, is introduced and used to prove and interrelate a number of different results. A compact subspace of a Hausdorff space is 𝜃-closed, and a 𝜃-closed subspace of a Hausdorff space is closed. A Hausdorff space X with property that every continuous function from X into a Hausdorff space is closed is shown to have the property that every 𝜃-continuous function from X into a Hausdorff space is closed. Those Hausdorff spaces in which the Fomin H-closed extension operator commutes with the projective cover (absolute) operator are characterized. An H-closed space is shown not to be the countable union of 𝜃-closed nowhere dense subspaces. Also, an equivalent form of Martin’s Axiom in terms of the class of H-closed spaces with the countable chain condition is given.

Mathematical Subject Classification 2000
Primary: 54D25
Received: 18 September 1974
Revised: 28 May 1975
Published: 1 August 1975
Raymond Frank Dickman
Jack Ray Porter