A quest for nice kernels of neighbourhood assignments

Raushan Z. Buzyakova; Vladimir Vladimirovich Tkachuk; Richard Gordon Wilson

Displaying similar documents to “A quest for nice kernels of neighbourhood assignments”

Diagonals and discrete subsets of squares

Dennis Burke, Vladimir Vladimirovich Tkachuk (2013)

Commentationes Mathematicae Universitatis Carolinae

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In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D \subset X \times X$ with $| D | = d (X)$ . We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $Σ$ -space $X$ and hence $X^{ω}$ is $d$ -separable. We give an example of a countably compact space $X$ such that $X^{ω}$ is not $d$ -separable. On the other hand, we show that for any Lindelöf $p$ -space $X$ there exists a discrete subset $D \subset X \times X$ such that $Δ = {(x, x) : x \in X} \subset \bar{D}$ ; in particular, the diagonal $Δ$ is a retract of...

Addition theorems, $D$ -spaces and dually discrete spaces

Ofelia Teresa Alas, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2009)

Commentationes Mathematicae Universitatis Carolinae

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A in a space $X$ is a family $𝒪 = {O_{x} : x \in X}$ of open subsets of $X$ such that $x \in O_{x}$ for any $x \in X$ . A set $Y \subseteq X$ is if $𝒪 (Y) = ⋃ {O_{x} : x \in Y} = X$ . If every neighbourhood assignment in $X$ has a closed and discrete (respectively, discrete) kernel, then $X$ is said to be a $D$ -space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf $P$ -space is a $D$ -space and we prove an addition theorem for metalindelöf...

A note on transitively $D$ -spaces

Liang-Xue Peng (2011)

Czechoslovak Mathematical Journal

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In this note, we show that if for any transitive neighborhood assignment $φ$ for $X$ there is a point-countable refinement $ℱ$ such that for any non-closed subset $A$ of $X$ there is some $V \in ℱ$ such that $| V \cap A | \geq ω$ , then $X$ is transitively $D$ . As a corollary, if $X$ is a sequential space and has a point-countable $w c s^{*}$ -network then $X$ is transitively $D$ , and hence if $X$ is a Hausdorff $k$ -space and has a point-countable $k$ -network, then $X$ is transitively $D$ . We prove that if $X$ is a countably compact sequential space and...

Convergence in compacta and linear Lindelöfness

Aleksander V. Arhangel&#039;skii, Raushan Z. Buzyakova (1998)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a compact Hausdorff space with a point $x$ such that $X ∖ {x}$ is linearly Lindelöf. Is then $X$ first countable at $x$ ? What if this is true for every $x$ in $X$ ? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when $X$ is, in addition, $ω$ -monolithic. We also prove that if $X$ is compact, Hausdorff, and $X ∖ {x}$ is strongly discretely Lindelöf, for every $x$ in $X$ , then $X$ is first countable. An example of linearly...

Displaying similar documents to “A quest for nice kernels of neighbourhood assignments”

Diagonals and discrete subsets of squares

Addition theorems, D -spaces and dually discrete spaces

A note on transitively D -spaces

Convergence in compacta and linear Lindelöfness

Addition theorems, $D$ -spaces and dually discrete spaces

A note on transitively $D$ -spaces