Initially κ-Compact and Related Spaces

Stephenson, R. M.

doi:10.1016/b978-0-444-86580-9.50016-1

Cited by 49 publications

(42 citation statements)

References 17 publications

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“…Our example is different since FR-spaces are initially m-chain compact, and it is known that for any m > oji the product of two (or even coi) initially m-chain compact spaces is initially m-compact [V2, Theorem 1]. Corollary 3.5 also answers consistently three questions raised by R. M. Stephenson, Jr., concerning products of totally initially m-compact spaces (see [St,pp. 624,625]).…”

Section: Theorem (Hechler [H]) If M Is a Countable Standard Model Ofmentioning

confidence: 60%

Sequentially compact, Franklin-Rajagopalan spaces

Nyikos¹,

Vaughan²

1987

Proc. Amer. Math. Soc.

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ABSTRACT. A locally compact T2-space is called a Franklin-Rajagopalan space (or FR-space) provided it has a countable discrete dense subset whose complement is homeomorphic to an ordinal with the order topology. We show that (1) every sequentially compact FR-space X can be identified with a space constructed from a tower T on w (X = X(T)), and (2) for an ultrafilter u on w, a sequentially compact FR-space X(T) is not u-compact if and only if there exists an ultrafilter uonu such that v D T, and v is below u in the RudinKeisler order on lj* . As one application of these results we show that in certain models of set theory there exists a family T of towers such that \T\ < 2W, and JT{X(T) : T € T} is a product of sequentially compact FR-spaces which is not countably compact (a new solution to the Scarborough-Stone problem). As further applications of these results, we give consistent answers to questions of van Douwen, Stephenson, and Vaughan concerning initially m-chain compact and totally initially m-compact spaces.

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Section: Theorem (Hechler [H]) If M Is a Countable Standard Model Ofmentioning

confidence: 60%

Sequentially compact, Franklin-Rajagopalan spaces

Nyikos¹,

Vaughan²

1987

Proc. Amer. Math. Soc.

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“…We get, as a special case, a classical result on products of "compact-like" spaces discussed in the introduction, compare [V1, Theorem 1] and [S,Theorem 5.4]. We note that its "midcompact case" seems to be new even in the framework of spaces.…”

Section: It Has Been Announced By D O L E C K I and G R E C O ([Dg Tmentioning

confidence: 93%

Infinite products of filters

Davis

Labuda

2007

Mathematica Slovaca

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“…Observe that if µ < λ(X), then each subset A ⊂ X of cardinality |A| ≤ µ has a complete accumulation point, so X is initially µ-compact (i.e., every open cover of X of size not bigger than µ has a finite subcover) [St,Theorem 2.2].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Lindelöf property and absolute embeddings

Bella

Yaschenko

1999

Proc. Amer. Math. Soc.

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Abstract. It is proved that a Tychonoff space is Lindelöf if and only if whenever a Tychonoff space Y contains two disjoint closed copies X 1 and X 2 of X, then these copies can be separated in Y by open sets. We also show that a Tychonoff space X is weakly C-embedded (relatively normal) in every larger Tychonoff space if and only if X is either almost compact or Lindelöf (normal almost compact or Lindelöf).

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Initially κ-Compact and Related Spaces

Cited by 49 publications

References 17 publications

Sequentially compact, Franklin-Rajagopalan spaces

Sequentially compact, Franklin-Rajagopalan spaces

Infinite products of filters

Lindelöf property and absolute embeddings

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