We solve a long standing question due to Arhangel'skii by constructing a compact space which has a Gδ cover with no continuum-sized (Gδ)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every Gδ cover has a -sized subcollection with a Gδ-dense union and that in a Lindelöf space with a base of multiplicity continuum, every Gδ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.

S. Spadaro, P. Szeptycki (2018). Gδ covers of compact spaces. ACTA MATHEMATICA HUNGARICA, 154(1), 252-263 [10.1007/s10474-017-0785-4].

Gδ covers of compact spaces

S. Spadaro
;
2018-01-01

Abstract

We solve a long standing question due to Arhangel'skii by constructing a compact space which has a Gδ cover with no continuum-sized (Gδ)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every Gδ cover has a -sized subcollection with a Gδ-dense union and that in a Lindelöf space with a base of multiplicity continuum, every Gδ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.
2018
Settore MAT/03 - Geometria
S. Spadaro, P. Szeptycki (2018). Gδ covers of compact spaces. ACTA MATHEMATICA HUNGARICA, 154(1), 252-263 [10.1007/s10474-017-0785-4].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/480994
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