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A new look at pointfree metrization theorems

Bernhard Banaschewski, Aleš Pultr (1998)

Commentationes Mathematicae Universitatis Carolinae

We present a unified treatment of pointfree metrization theorems based on an analysis of special properties of bases. It essentially covers all the facts concerning metrization from Engelking [1] which make pointfree sense. With one exception, where the generalization is shown to be false, all the theorems extend to the general pointfree context.

A new metrization theorem

F. G. Arenas, M. A. Sánchez-Granero (2002)

Bollettino dell'Unione Matematica Italiana

We give a new metrization theorem on terms of a new structure introduced by the authors in [2] and called fractal structure. As a Corollary we obtain Nagata-Smirnov’s and Uryshon’s metrization Theorems.

A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees

Akira Iwasa, Peter J. Nyikos (2006)

Commentationes Mathematicae Universitatis Carolinae

It is independent of the usual (ZFC) axioms of set theory whether every collectionwise Hausdorff tree is either metrizable or has an uncountable chain. We show that even if we add “or has an Aronszajn subtree,” the statement remains ZFC-independent. This is done by constructing a tree as in the title, using the set-theoretic hypothesis ♦ * , which holds in Gödel’s Constructible Universe.

A nontransitive space based on combinatorics

Hans-Peter A. Künzi, Stephen Watson (1999)

Bollettino dell'Unione Matematica Italiana

Costruiamo uno spazio nontransitivo analogo al piano di Kofner. Mentre gli argomenti usati per la costruzione del piano di Kofner si fondano su riflessioni geometriche, le nostre prove si basano su idee combinatorie.

A note on g -metrizable spaces

Jinjin Li (2003)

Czechoslovak Mathematical Journal

In this paper, the relationships between metric spaces and g -metrizable spaces are established in terms of certain quotient mappings, which is an answer to Alexandroff’s problems.

A note on operators extending partial ultrametrics

Edward D. Tymchatyn, Michael M. Zarichnyi (2005)

Commentationes Mathematicae Universitatis Carolinae

We consider the question of simultaneous extension of partial ultrametrics, i.e. continuous ultrametrics defined on nonempty closed subsets of a compact zero-dimensional metrizable space. The main result states that there exists a continuous extension operator that preserves the maximum operation. This extension can also be chosen so that it preserves the Assouad dimension.

A note on ℵ -spaces and g -metrizable spaces

Zhaowen Li (2005)

Czechoslovak Mathematical Journal

In this paper, we give the mapping theorems on ℵ -spaces and g -metrizable spaces by means of some sequence-covering mappings, mssc-mappings and π -mappings.

A note on star Lindelöf, first countable and normal spaces

Wei-Feng Xuan (2017)

Mathematica Bohemica

A topological space X is said to be star Lindelöf if for any open cover 𝒰 of X there is a Lindelöf subspace A ⊂ X such that St ( A , 𝒰 ) = X . The “extent” e ( X ) of X is the supremum of the cardinalities of closed discrete subsets of X . We prove that under V = L every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under MA + ¬ CH , which shows that a star Lindelöf, first countable and normal space may not have countable extent.

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