Countably Compact Spaces and Martin's Axiom

1978 ◽  
Vol 30 (02) ◽  
pp. 243-249 ◽  
Author(s):  
William Weiss
Keyword(s):  
Set Theory ◽  
Normal Space ◽  
Topological Spaces ◽  
Compact Spaces ◽  
Countably Compact Space ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Perfectly Normal Space ◽  
The Relationship ◽  
Perfectly Normal

The relationship between compact and countably compact topological spaces has been studied by many topologists. In particular an important question is: “What conditions will make a countably compact space compact?” Conditions which are “covering axioms” have been extensively studied. The best results of this type appear in [19]. We wish to examine countably compact spaces which are separable or perfectly normal. Recall that a space is perfect if and only if every closed subset is a Gδ, and that a space is perfectly normal if and only if it is both perfect and normal. We show that the following statement follows from MA +┐ CH and thus is consistent with the usual axioms of set theory: Every countably compact perfectly normal space is compact. This result is Theorem 3 and can be understood without reading much of what goes before.

scholarly journals Characterizations of Strongly Paracompact Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Xin Zhang
Keyword(s):  
Open Cover ◽  
Normal Space ◽  
Compact Spaces ◽  
Countably Paracompact ◽  
Star Countable ◽  
Perfectly Normal Space ◽  
Paracompact Spaces ◽  
Perfectly Normal

Characterizations of strongly compact spaces are given based on the existence of a star-countable open refinement for every increasing open cover. It is proved that a countably paracompact normal space (a perfectly normal space or a monotonically normal space) is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement. Moreover, it is shown that a space is linearlyDprovided that every increasing open cover of the space has a point-countable open refinement.

Locally Compact Normal Spaces in the Constructible Universe

1982 ◽  
Vol 34 (5) ◽  
pp. 1091-1096 ◽  
Author(s):  
W. Stephen Watson
Keyword(s):  
Set Theory ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces ◽  
Perfectly Normal

Arhangel'skiĭ proved around 1959 [1] that, for the class of perfectly normal locally compact spaces, metacompactness and paracompactness are equivalent. It is shown to be consistent that this equivalence holds for the (larger) class of normal locally compact spaces (answering a question of Tall [8], [9]).The consistency of the existence of locally compact normal noncollectionwise Hausdorff spaces has been known since 1967. It is shown that the existence of such spaces is independent of the axioms of set theory, thus establishing that Bing's example G cannot be modified under ZFC to be locally compact.All topological spaces are assumed to be Hausdorff.First, a definition and three standard lemmata are needed.

Selections and countable compactness

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
David Buhagiar ◽  
Valentin Gutev
Keyword(s):  
Compact Spaces ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Countable Compactness

AbstractThe present paper deals with continuous extreme-like selections for the Vietoris hyperspace of countably compact spaces. Several new results and applications are established, along with some known results which are obtained under minimal hypotheses. The paper contains also a number of examples clarifying the role of countable compactness.

scholarly journals Minimal sequential Hausdorff spaces

2004 ◽  
Vol 2004 (22) ◽  
pp. 1169-1177
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Hausdorff Spaces ◽  
Sequential Space ◽  
Compact Spaces ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Sequential Spaces ◽  
By Products

A sequential space(X,T)is called minimal sequential if no sequential topology onXis strictly weaker thanT. This paper begins the study of minimal sequential Hausdorff spaces. Characterizations of minimal sequential Hausdorff spaces are obtained using filter bases, sequences, and functions satisfying certain graph conditions. Relationships between this class of spaces and other classes of spaces, for example, minimal Hausdorff spaces, countably compact spaces, H-closed spaces, SQ-closed spaces, and subspaces of minimal sequential spaces, are investigated. While the property of being sequential is not (in general) preserved by products, some information is provided on the question of when the product of minimal sequential spaces is minimal sequential.

scholarly journals PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 210-229
Author(s):  
O. Maslyuchenko ◽  
A. Kushnir
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Normal Space ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Pseudocompact Space ◽  
Upper Semicontinuous Function ◽  
Perfectly Normal Space ◽  
Perfectly Normal

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

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