Vol. 44, No. 2, 1973

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On Hausdorff compactifications

Marlon C. Rayburn

Vol. 44 (1973), No. 2, 707–714

Given a pair of spaces X and Y , a necessary and sufficient condition is found for Y to be homeomorphic to cl αX(αX X) for some compactification αX of X. From this follows a necessary and sufficient condition for Y to be homeomorphic to αX X for some αX. As an application, a sufficient condition is found to insure the isomorphism of the upper semi-lattices of compactifications K(X) and K(Y ) for arbitrary X and Y , and in consequence it appears that for every space X, there is a pseudocompact space Y with K(X) isomorphic to K(Y ). A necessary condition for K(X) to be isomorphic to K(Y ) is observed for arbitrary X and Y , and this leads to the consideration of spaces compactly generated at infinity. Examples are constructed.

Mathematical Subject Classification 2000
Primary: 54D35
Secondary: 54D50
Received: 30 September 1971
Revised: 27 September 1972
Published: 1 February 1973
Marlon C. Rayburn