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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 fibered approximation property in dimension n

Taras Banakh, Vesko Valov (2010)

Open Mathematics

A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: $𝕀$ m × $𝕀$ n → M there exists a map g′: $𝕀$ m × $𝕀$ n → M such that g′ is ɛ-homotopic to g and dim g′ (z × $𝕀$ n) ≤ n for all z ∈ $𝕀$ m. The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].

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 regular $G_{δ}$ -diagonals

Zenor, P. (1972)

General Topology and its Relations to Modern Analysis and Algebra

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 $σ$ -locally countable weak-bases

Zhaowen Li (2006)

Archivum Mathematicum

In this paper, spaces with $σ$ -locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or $g$ -first countable spaces with $σ$ -locally countable $c s$ -networks).

Spaces With $σ$ -Locally Finite Lindelöf sn-Networks

Luong Quoc Tuyen (2013)

Publications de l'Institut Mathématique

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.

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