Vol. 105, No. 2, 1983

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The pinched-cube topology

Zbigniew Piotrowski, A. Rosłanowski and Brian M. Scott

Vol. 105 (1983), No. 2, 399–413

McCoy introduced a topology intermediate between the Tikhonov and box product topologies on the countably infinite power ωX of a topological space X. He used this topology to study Baire category in 2X, the hyperspace of closed subsets of X in the Vietoris topology. In this note we generalize this ‘pinched-cube’ topology to arbitrary infinite powers, κX, of X and investigate the extent to which it inherits fundamental properties of X.

In §1 we introduce basic definitions and elementary facts. Separation axioms are considered in §2, compactness, connectedness, and separability in §3. In §4 we consider some completeness properties, and in §5 we explicate the connection with hyperspaces.

Mathematical Subject Classification 2000
Primary: 54B10
Secondary: 54D10, 54D99
Received: 5 August 1981
Revised: 31 August 1981
Published: 1 April 1983
Zbigniew Piotrowski
A. Rosłanowski
Brian M. Scott