scholarly journals Maximal pseudocompact spaces and the Preiss-Simon property

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Ofelia Alas ◽  
Vladimir Tkachuk ◽  
Richard Wilson
Keyword(s):  
Compact Space ◽  
Compact Spaces ◽  
Pseudocompact Space ◽  
Pseudocompact Spaces ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Continuous Images

AbstractWe study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.

scholarly journals F-points in countably compact spaces

2001 ◽  
Vol 2 (1) ◽  
pp. 33 ◽  
Author(s):  
Angelo Bella ◽  
V.I. Malykhin
Keyword(s):  
Compact Space ◽  
Compact Spaces ◽  
Extremally Disconnected ◽  
Countable Tightness ◽  
Countably Compact ◽  
Countably Compact Spaces

Answering a question of A.V. Arhangel'skii, we show that any extremally disconnected subspace of a compact space with countable tightness is discrete.

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.

Continuous Images of Compact Semilattices

1987 ◽  
Vol 30 (1) ◽  
pp. 109-113 ◽  
Author(s):  
Murray Bell ◽  
Jan Pelant
Keyword(s):  
Compact Space ◽  
Infinite Cardinal ◽  
Theoretic Property ◽  
Compact Spaces ◽  
General Order ◽  
Isolated Point ◽  
Continuous Images

AbstractHyadic spaces are the continuous images of a hyperspace of a compact space. We prove that every non-isolated point in a hyadic space is the endpoint of some infinite cardinal subspace. We isolate a more general order-theoretic property of hyerspaces of compact spaces which is also enjoyed by compact semilattices from which the theorem follows.

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