Abstract
A Tychonoff space X is called strongly Čech-complete if there exist paracompact open subsets V1,V2,… of βX such that ⋂n=1∞Vn=X. Strong Čech-completeness of a Tychonoff space is characterized in terms of the existence of certain kinds of complete sequence of open covers, for example, of a complete sequence consisting of star-finite open covers. A metrizable space is shown to be strongly Čech-complete if, and only if, the space is Čech-complete and strongly metrizable. Universal spaces for strongly Čech-complete metrizable spaces are indicated.A compatible complete metric is constructed for R such that, for each r>0, every open r-ball meets at most 25 distinct r-balls. This metric is used to derive characterizations for strongly Čech-complete metrizable spaces and strongly metrizable spaces in terms of special compatible metrics.
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