Vol. 37, No. 1, 1971

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On the open continuous images of paracompact Čech complete spaces

Howard Henry Wicke and John Mays Worrell Jr.

Vol. 37 (1971), No. 1, 265–275

This article characterizes the completely regular T0 open continuous images of paracompact Čech complete spaces. The characterization involves three conditions equivalent to being such an image. The first is an intrinsic condition concerning the position of the space in any of its Hausdorff bicompactifications. This condition weakens the condition of Čech completeness by replacing the concept of Gδ-set by that of set of interior condensation. This replacement yields a notion of topological completeness which has certain advantages over Čech completeness and uniform completeness but which reduces to Cech completeness in the case of metrizable spaces. The second comdition (Condition 𝒦) is intrinsically defined with the use of a sequence of collections of open sets. It is an analogue of the notion of a regular T0-space having a monotonically complete base of countable order. The third condition is that of being an open continuous image of a space which is the sum of open Čech complete subspaces. The main theorem thus displays four equivalent forms of a topological completeness property invariant under open continuous mappings between Tychonoff spaces.

Mathematical Subject Classification 2000
Primary: 54D20
Received: 3 July 1968
Published: 1 April 1971
Howard Henry Wicke
John Mays Worrell Jr.