Murray G. Bell. Spaces of ideals of partial functions. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 1–4. - Alan Dow. Compact spaces of countable tightness in the Cohen model. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 55–67. - Peter J. Nyikos. Classes of compact sequential spaces. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 135–159. - Franklin D. Tall. Topological problems for set-theorists. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 194–200.

1991 ◽  
Vol 56 (2) ◽  
pp. 753-755
Author(s):  
Judith Roitman
Keyword(s):  
Set Theory ◽  
Partial Functions ◽  
Compact Spaces ◽  
Countable Tightness ◽  
Lecture Notes ◽  
Model Set ◽  
Cohen Model ◽  
Sequential Spaces

Products of some special compact spaces and restricted forms of AC

2010 ◽  
Vol 75 (3) ◽  
pp. 996-1006 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis
Keyword(s):  
Set Theory ◽  
Closed Subset ◽  
Choice Function ◽  
Ordinal Number ◽  
Axiom Of Choice ◽  
Finite Support ◽  
Compact Spaces ◽  
The Axiom Of Choice ◽  
Infinite Set

AbstractWe establish the following results:1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

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.

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.

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.

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