Some properties related to [a,b]-compactness

Vaughan, Jerry E.

doi:10.4064/fm-87-3-251-260

Cited by 21 publications

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“…Notice that [µ, λ]-compactness is a notion which encompasses both Lindelöfness (more generally, κ-final compactness) and countable compactness (more generally, κ-initial compactness). See, e. g., [Ca2,Gá,Li1,Li2,Va] and references there for further information about [µ, λ]compactness.…”

Section: A Two Cardinal Generalization Of Pseudocompactnessmentioning

confidence: 99%

More generalizations of pseudocompactness

Lipparini¹

2011

Topology and its Applications

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We introduce a covering notion depending on two cardinals, which we call O - [ μ, λ ] -\ud compactness, and which encompasses both pseudocompactness and many other known\ud generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out\ud to be equivalent to O - [ ω, ω ] -compactness.\ud We provide several characterizations of O - [ μ, λ ] -compactness, and we discuss its connec-\ud tion with D-pseudocompactness, for D an ultrafilter. The connection turns out to be rather\ud strict when the above notions are considered with respect to products. In passing, we pro-\ud vide some conditions equivalent to D-pseudocompactness.\ud Finally, we show that our methods provide a unified treatment both for O - [ μ, λ ] -\ud compactness and for [ μ, λ ] -compactness

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Section: A Two Cardinal Generalization Of Pseudocompactnessmentioning

confidence: 99%

More generalizations of pseudocompactness

Lipparini¹

2011

Topology and its Applications

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“…The assumption that κ is regular is necessary in Proposition 11: see [20] for various counterexamples.…”

Section: (I) a Topological Space Is Finally κ-Compact If And Only If mentioning

confidence: 99%

Compact factors in finally compact products of topological spaces

Lipparini

2006

Topology and its Applications

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We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors. The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelof then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom

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“…It has been studied by many people, sometimes under different names and notations, and in several equivalent formulations. See a survey of further related notions and results in [34]. See also, e. g., [10,20,33] and references there for further information.…”

Section: Rµ λS-compactness (For Cardinals)mentioning

confidence: 99%

“…This fact has found many applications, mainly due to the fact that, for ν regular, rν, νs-compactness has many equivalent formulations, most notably in terms of the existence of complete accumulation points of sets of cardinality ν. See [1,33,34]. See also [25,Section 6] and [18], where the above mentioned "slicing" procedure has found other substantial applications.…”

Section: Rµ λS-compactness (For Cardinals)mentioning

confidence: 99%

Ordinal compactness

Lipparini

2020

Filomat

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We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities. The most general form depends on two ordinal parameters. Ordinal compactness turns out to be a much more varied notion than cardinal compactness. We prove many nontrivial results of the form ?every [?,?]-compact topological space is [?',?']-compact?, for ordinals ?,?, ?'and ?' while only trivial results of the above form hold, if we restrict to regular cardinals. Counterexamples are provided showing that many results are optimal. Many spaces satisfy the very same cardinal compactness properties, but have a broad range of distinct behaviors, as far as ordinal compactness is concerned. A much more refined theory is obtained for T1 spaces, in comparison with arbitrary topological spaces. The notion of ordinal compactness becomes partly trivial for spaces of small cardinality.

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Some properties related to [a,b]-compactness

Cited by 21 publications

References 0 publications

More generalizations of pseudocompactness

More generalizations of pseudocompactness

Compact factors in finally compact products of topological spaces

Ordinal compactness

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