Hereditary separability in Hausdorff continua

Daniel, D.; Tuncali, Murat

doi:10.4995/agt.2012.1638

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…It is known (see [17], [19]) that each monotonically normal space X is sub-hereditarily separable in the sense that each separable subspace of X is hereditarily separable. Sub-hereditarily separable spaces were introduced and studied by D. Daniel and M. Tuncali in [6]. The sub-hereditary separability of monotonically normal spaces and Theorem 2.4 motivate the following: Problem 2.5.…”

Section: Some Properties Of Hereditarily Supercompact Spacesmentioning

confidence: 99%

Hereditarily supercompact spaces

Banakh,

Kosztolowicz,

Turek

2013

Preprint

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A topological space X is called hereditarily supercompact if each closed subspace of X is supercompact. By a combined result of Bula, Nikiel, Tuncali, Tymchatyn, and Rudin, each monotonically normal compact Hausdorff space is hereditarily supercompact. A dyadic compact space is hereditarily supercompact if and only if it is metrizable. Under (MA+¬CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindelöf. This implies that under (MA+¬CH) a scattered compact space is metrizable if and only if it is separable and hereditarily supercompact. The hereditary supercompactness is not productive: the product [0, 1] × αD of the closed interval and the one-point compactification αD of a discrete space D of cardinality |D| ≥ non(M) is not hereditarily supercompact (but is Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption cof(M) = ω1 the space [0, 1] × αD contains a closed subspace X which is first countable and hereditarily paracompact but not supercompact.

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Section: Some Properties Of Hereditarily Supercompact Spacesmentioning

confidence: 99%

Hereditarily supercompact spaces

Banakh,

Kosztolowicz,

Turek

2013

Preprint

View full text Add to dashboard Cite

show abstract

Hereditarily supercompact spaces

Банах

Kosztołowicz

Turek

2014

Topology and its Applications

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A topological space $X$ is called hereditarily supercompact if each closed subspace of X is supercompact. By a combined result of Bula, Nikiel, Tuncali, Tymchatyn, and Rudin, each monotonically normal compact Hausdorff space is hereditarily supercompact. A dyadic compact space is hereditarily supercompact if and only if it is metrizable. Under (MA + not CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindel\"of. This implies that under (MA + not CH) a scattered compact space is metrizable if and only if it is separable and hereditarily supercompact. The hereditary supercompactness is not productive: the product [0,1] x \alpha D of the closed interval and the one-point compactification \alpha D of a discrete space D of cardinality |D|\ge non(M) is not hereditarily supercompact (but is Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption cof(M)=\omega_1 the space [0,1] x \alpha D contains a closed subspace X which is first countable and hereditarily paracompact but not supercompact.Comment: 12 page

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Hereditary separability in Hausdorff continua

Cited by 2 publications

References 17 publications

Hereditarily supercompact spaces

Hereditarily supercompact spaces

Hereditarily supercompact spaces

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