OpenUCT :: Browsing by Author "Flax, Cyril Lee"

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Syntopogenous structures and real-compactness
(1972) Flax, Cyril Lee; Brümmer, Guillaume C L
The syntopogenous structures were introduced by Á. Császár. These are generalisations of classical continuity structures such as topologies, proximities and uniformities. In his book, Foundations of General Topology (1963) (Preceded by a French (1960) and a German (1963) edition), Császár treated many properties of syntopgenous structures. Among these properties were completeness and compactness, but not realcompactness. Our purpose was to extend the definition of realcompactness from uniformisable topologies to arbitrary syntopogenous structures and to produce a real compact reflection for arbitrary syntopogenous structures. We did not fully accomplish this purpose. We have, in fact, first defined a notion of quasirealcompactness for arbitrary syntopogenous structures. For uniformisable Hausdorff topologies, realcompactness implies quasirealcompactness; we could not prove or disprove the converse implication. Nevertheless, we were able to give a characterisation of realcompactness for a uniformisable Hausdorff topology in terms of quasirealcompactness of a certain induced proximity; moreover, we produced a double quasirealcompact reflection in the category of separated syntopogenous structures, and from this retrieved the classical Hewitt realcompact reflection.