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2012, vol. 64, br. 2, str. 125-137
On certain separable and connected refinements of the Euclidean topology
(naslov ne postoji na srpskom)
Institute of Mathematics, University of Natural Resources and Life Sciences, Vienna, Austria

Ključne reči: σ -ideal; Lebesgue null set; meager; separable; totally pathwise disconnected
Sažetak
(ne postoji na srpskom)
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 U I is dense and Q ∩UI=0. Then for each I ∈ Σ the family η/I of all sets X \Y with X ∈ η and Y ∈ I is a topology on R. Such a refinement of η always preserves separability 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 set and 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 I consists of either all countable sets or all null sets or all meager sets or all sets contained in R\Q. Finally we track down a subfamily Σ1 of Σ with cardinality 2²c such that (R, η/I) and (R, n/J) are never homeomorphic for distinct I, J, ;J in Σ1.
 Reference Brendle, J. (1991) Larger cardinals in Cichon's diagram. J. Symb. Logic, (56): 795-810 Comfort, W.W., Negrepontis, S. (1974) The theory of ultrafilters. Springer Janković, D., Hamlett, T.R. (1990) New topologies from old via ideals. Amer. Math. Monthly, 97 (4), 295-310 Kuratowski, K. (1966) Topology. New York: Academic Press, I Martin, N.F.G. (1961) Generalized condensation points. Duke Mathematical Journal, 28(4): 507-514 Oxtoby, J.C. (1980) Measure and category. New York: Springer-Verlag Samuels, P. (1975) A topology formed from a given topology and ideal. J London Math. Soc, 2, 10, 409-416 Scheinberg, S. (1971) Topologies which generate a complete measure algebra. Advances in Mathematics, 7(3): 231-239 Steen, L.A., Seebach, J.A. (1995) Counterexamples in topology. Dover