scholarly journals On the problem of condensation onto compacta

2019 ◽  
Vol 488 (2) ◽  
pp. 130-132
Author(s):  
A. V. Osipov ◽  
E. G. Pytkeev
Keyword(s):  
Topological Space ◽  
Continuum Hypothesis ◽  
Compact Topological Space ◽  
The Continuum ◽  
Perfectly Normal

The paper proves (assuming the continuum hypothesis CH) that there exists a perfectly normal compact topological space Z and a countable set E ⊂ Z, such that Z\E is not condensed onto a compact. The existence of such a space answers (in CH) negatively to the question of V.I. Ponomareva: Is every perfectly normal compact an α-space? It is proved that in the class of ordered compacts the property of being an α-space is not multiplicative.

Uncountable Discrete Sets in Extensions and Metrizability

1982 ◽  
Vol 25 (4) ◽  
pp. 472-477 ◽  
Author(s):  
Murray Bell ◽  
John Ginsburg
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuum Hypothesis ◽  
Vietoris Topology ◽  
Discrete Subset ◽  
Similar Argument ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

AbstractIf X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

scholarly journals A generalization of a theorem of Aquaro

1973 ◽  
Vol 9 (1) ◽  
pp. 105-108 ◽  
Author(s):  
C.E. Aull
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuum Hypothesis ◽  
Open Cover ◽  
Moore Space ◽  
Countably Compact Space ◽  
Classic Result ◽  
Countably Compact ◽  
The Continuum

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

scholarly journals Quasi-normality of Mrowka spaces

2020 ◽  
Vol 13 (3) ◽  
pp. 697-700
Author(s):  
Ibtesam Eid AlShammari ◽  
Lutfi Kalantan
Keyword(s):  
Topological Space ◽  
Continuum Hypothesis ◽  
The Continuum

A topological space X is called quasi-normal if X is regular and any two disjoint π-closed subsets A and B of X are separated. We give a Mr ́owka space which is not quasi-normal and use the continuum hypothesis (CH) and truly cardinality c to present Mr ́owka spaces which are quasi-normal.

scholarly journals Martin's axiom and a regular topological space with uncountable net weight whose countable product is hereditarily separable and hereditarily Lindelöf

1987 ◽  
Vol 52 (2) ◽  
pp. 396-399
Author(s):  
Krzysztof Ciesielski
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Negative Answer ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countable Product ◽  
Generic Filter ◽  
Cohen Forcing ◽  
The Continuum

In [1, p. 51] A. V. Arhangel'skiĭ, in connection with the problems of L-spaces and S-spaces, examined further the notions of hereditary separability and hereditary Lindelöfness. In particular he considered the following property P: “Every regular topological space has a countable net weight provided its countable product is hereditarily Lindelöf and hereditarily separable.” He noticed that the continuum hypothesis implies negation of the property P and posed a question: “Do Martin's Axiom and the negation of the continuum hypothesis imply P?” The purpose of this paper is to give a negative answer to this question.The set-theoretical and topological notation that we use is standard and can be found in [6] and [5] respectively.Throughout the paper we will use the notation H(X, Y) to denote the set of all finite functions from a set X to Y.Theorem. Con(ZFC) → Con(ZFC + MA + ¬CH + there exists a 0-dimensional Hausdorff space X such that nw(X) = с and nw(Y) = ω for any Y ϵ [X]<с).Proof. Let M be a model of ZFC satisfying CH and let F be an M-generic filter over the Cohen forcing {H(ω2 × ω2, 2), ⊃). Then f = ⋃F is a function and f: ω2 × ω2 → 2.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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