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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.

An observation on spaces with a zeroset diagonal

Wei-Feng Xuan — 2020

Mathematica Bohemica

We say that a space X has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. A space X has a zeroset diagonal if there is a continuous mapping f : X 2 [ 0 , 1 ] with Δ X = f - 1 ( 0 ) , where Δ X = { ( x , x ) : x X } . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most 𝔠 .

Cardinalities of DCCC normal spaces with a rank 2-diagonal

Wei-Feng XuanWei-Xue Shi — 2016

Mathematica Bohemica

A topological space X has a rank 2-diagonal if there exists a diagonal sequence on X of rank 2 , that is, there is a countable family { 𝒰 n : n ω } of open covers of X such that for each x X , { x } = { St 2 ( x , 𝒰 n ) : n ω } . We say that a space X satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. We mainly prove that if X is a DCCC normal space with a rank 2-diagonal, then the cardinality of X is at most 𝔠 . Moreover, we prove that if X is a first countable...

Spaces with property ( D C ( ω 1 ) )

Wei-Feng XuanWei-Xue Shi — 2017

Commentationes Mathematicae Universitatis Carolinae

We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

Some results on spaces with 1 -calibre

Wei-Feng XuanWei-Xue Shi — 2016

Commentationes Mathematicae Universitatis Carolinae

We prove that, assuming , if X is a space with 1 -calibre and a zeroset diagonal, then X is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular G δ -diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with 1 -calibre.

Some results on semi-stratifiable spaces

Wei-Feng XuanYan-Kui Song — 2019

Mathematica Bohemica

We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If X is a semi-stratifiable space, then X is separable if and only if X is D C ( ω 1 ) ; (2) If X is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then X is separable; (3) Let X be a ω -monolithic star countable extent semi-stratifiable space. If t ( X ) = ω and d ( X ) ω 1 , then X is hereditarily separable. Finally, we prove that for any T 1 -space...

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