Authors
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Gerald Kuba
Institute of Mathematics, University of Natural Resources and Life Sciences, Vienna, Austria
Author
Keywords:
-ideal, Lebesgue null set, meager, separable, totally pathwise disconnected
Abstract
Write for the cardinality of the continuum and let be the Euclidean topology on .Let be the family of all -ideals on such that is dense and . Then for each the family of all sets with and is a topology on . Such a refinement of always preservesseparability and connectedness, but destroys metrizability (and first countability almost always)and makes the space totally pathwise disconnected. Nevertheless, the separable Hausdorff space still has the two metric properties that every point is reachable by a sequence of points within any fixed countable dense setand that (even in the absence of first countability) sequential continuity is strong enough to entail continuity.In detail we investigate further main properties in the four most interesting cases when the -ideal consists of either all countable sets or all null sets or all meager sets or all sets contained in .Finally we track down a subfamily of with cardinality such that and are never homeomorphic for distinct in .