On certain separable and connected refinements of the Euclidean topology

Authors

  • Gerald Kuba Institute of Mathematics, University of Natural Resources and Life Sciences, Vienna, Austria Author

Keywords:

σ-ideal, Lebesgue null set, meager, separable, totally pathwise disconnected

Subjects:

54G20, 54A10

Abstract

Write c for the cardinality of the continuum and let η be the Euclidean topology on R.Let Σ be the family of all σ-ideals I on R such that I is dense and QI=. Then for each IınΣ the family η/I of all setsXY with Xınη and YınI is a topology on R. Such a refinement of η always preservesseparability and connectedness, but destroys metrizability (and first countability almost always)and makes the space totally pathwise disconnected. Nevertheless, the separable Hausdorff space (R,η/I)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 σ-ideal Iconsists of either all countable sets or all null sets or all meager sets or all sets contained in RQ.Finally we track down a subfamily Σ1 of Σ with cardinality 22c such that (R,η/I) and (R,η/J) are never homeomorphic for distinct I,J in Σ1.

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Published

2012-04-15