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.
