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In this paper, we prove that a space $X$ is a sequentially-quotient $π$ -image of a metric space if and only if $X$ has a point-star $s n$ -network consisting of $c s^{*}$ -covers. By this result, we prove that a space $X$ is a sequentially-quotient $π$ -image of a separable metric space if and only if $X$ has a countable $s n$ -network, if and only if $X$ is a sequentially-quotient compact image of a separable metric space; this answers a question raised by Shou Lin affirmatively. We also obtain some results on spaces with countable...

Spaces with increment of dimension n

M. Charalambous (1976)

Fundamenta Mathematicae

Spaces with large relative extent

Yan-Kui Song (2007)

Czechoslovak Mathematical Journal

In this paper, we prove the following statements: (1) For every regular uncountable cardinal $κ$ , there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e (Y, X) \geq κ$ , but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf...

Spaces with large star cardinal number

Yan-Kui Song (2012)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we prove the following statements: (1) For any cardinal $κ$ , there exists a Tychonoff star-Lindelöf space $X$ such that $a (X) \geq κ$ . (2) There is a Tychonoff discretely star-Lindelöf space $X$ such that $a a (X)$ does not exist. (3) For any cardinal $κ$ , there exists a Tychonoff pseudocompact $σ$ -starcompact space $X$ such that $st - l (X) \geq κ$ .

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-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 $𝔠$ .

Spaces with star countable extent

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

Commentationes Mathematicae Universitatis Carolinae

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 $A \subset X$ 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 space with...

Spaces with zero set bases

Michael L. Wage (1977)

Commentationes Mathematicae Universitatis Carolinae

Spaces with $σ$ - $n$ -linked topologies as special subspaces of separable spaces

Ronnie Levy, Mikhail Matveev (1999)

Commentationes Mathematicae Universitatis Carolinae

We characterize spaces with $σ$ - $n$ -linked bases as specially embedded subspaces of separable spaces, and derive some corollaries, such as the $𝐜$ -productivity of the property of having a $σ$ -linked base.

Spaces with $σ$ -weakly hereditarily closure-preserving sn-networks.

Ge, Xun, Shen, Jianhua, Ying, Ge (2007)

Novi Sad Journal of Mathematics

Spaces without Large Projective Subspaces.

J.R. Isbell (1965)

Mathematica Scandinavica

Spaces Y Homeomorphic to ßY Y.

W. Wistar Comfort, Liam O'Callaghan (1978)

Mathematische Zeitschrift

Spaces $α$ - $s g$ - $T_{i}$ $i = 0, 1, 2, 3$ .

Rosas, Ennis (2003)

Divulgaciones Matemáticas

$Spec (R)$ and separation axioms between $T_{0}$ and $T_{1}$ .

Ávila G., Jesús Antonio (2005)

Divulgaciones Matemáticas

Special bases for compact metrizable spaces

Eric van Douwen (1981)

Fundamenta Mathematicae

Special lattices of compactifications

Alessandro Caterino (1985)

Commentationes Mathematicae Universitatis Carolinae

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

We recall some classical results relating normality and some natural weakenings of normality in $Ψ$ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$ -sets, $λ$ -sets and $σ$ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $ℵ_{0}$ -separated. This new class fits between $λ$ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being potentially...

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