Some versions of second countability of metric spaces in ZF and their role to compactness

Kyriakos Keremedis

Displaying similar documents to “Some versions of second countability of metric spaces in ZF and their role to compactness”

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...

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...

When is $𝐍$ Lindelöf?

Horst Herrlich, George E. Strecker (1997)

Commentationes Mathematicae Universitatis Carolinae

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Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $ℕ$ is a Lindelöf space, (2) $ℚ$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$ , a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$ , (7) a function $f : ℝ \to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous...

Three small results on normal first countable linearly H-closed spaces

Mathieu Baillif (2022)

Commentationes Mathematicae Universitatis Carolinae

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We use topological consequences of , and proved by other authors to show that normal first countable linearly H-closed spaces with various additional properties are compact in these models.

Weak-bases and $D$ -spaces

Dennis K. Burke (2007)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that certain weak-base structures on a topological space give a $D$ -space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are $D$ -spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Hence, quotient mappings, with compact fibers, from metric spaces have a $D$ -space image. What about quotient $s$ -mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base...

Displaying similar documents to “Some versions of second countability of metric spaces in ZF and their role to compactness”

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

On the metric reflection of a pseudometric space in ZF

When is 𝐍 Lindelöf?

Three small results on normal first countable linearly H-closed spaces

Weak-bases and D -spaces

When is $𝐍$ Lindelöf?

Weak-bases and $D$ -spaces