Displaying similar documents to “Weak-bases and D -spaces”

The sup = max problem for the extent and the Lindelöf degree of generalized metric spaces, II

Yasushi Hirata (2015)

Commentationes Mathematicae Universitatis Carolinae

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In [The sup = max problem for the extent of generalized metric spaces, Comment. Math. Univ. Carolin. The special issue devoted to Čech 54 (2013), no. 2, 245–257], the author and Yajima discussed the sup = max problem for the extent and the Lindelöf degree of generalized metric spaces: (strict) p -spaces, (strong) Σ -spaces and semi-stratifiable spaces. In this paper, the sup = max problem for the Lindelöf degree of spaces having G δ -diagonals and for the extent of spaces having point-countable...

Spaces with star countable extent

A. D. Rojas-Sánchez, Angel Tamariz-Mascarúa (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a topological property P , we say that a space X is star P if for every open cover 𝒰 of the space X there exists A ⊂ X such that s t ( A , 𝒰 ) = X . We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf...

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

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Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class ₂ of absolute G δ σ -sets; moreover, if Y is an absolute F σ δ -set, then X ω contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute G δ -set, then X ω contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable...

On the metric reflection of a pseudometric space in ZF

Horst Herrlich, Kyriakos Keremedis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show: (i) The countable axiom of choice 𝐂𝐀𝐂 is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom 𝐂𝐌𝐂 is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice 𝐀𝐂 is equivalent to each...

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space M D the following four conditions are equivalent: (i) K is fragmented by d D , where, for each S ⊂ D, d S ( x , y ) = s u p ϱ ( x ( t ) , y ( t ) ) : t ∈ S . (ii) For each countable subset...

Addition theorems for dense subspaces

Aleksander V. Arhangel'skii (2015)

Commentationes Mathematicae Universitatis Carolinae

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We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space X which is the union of two dense metrizable subspaces need not be a p -space. However, if a normal space X is the union of a finite family μ of dense subspaces each of which is metrizable by a complete metric, then X is also metrizable...