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1
Content available remote

Monotone weak Lindelöfness

100%
EN
The definition of monotone weak Lindelöfness is similar to monotone versions of other covering properties: X is monotonically weakly Lindelöf if there is an operator r that assigns to every open cover U a family of open sets r(U) so that (1) ∪r(U) is dense in X, (2) r(U) refines U, and (3) r(U) refines r(V) whenever U refines V. Some examples and counterexamples of monotonically weakly Lindelöf spaces are given and some basic properties such as the behavior with respect to products and subspaces are discussed.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1635-1642
EN
We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of x, then X is metacompact.
3
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A note on the extent of two subclasses of star countable spaces

81%
Open Mathematics
|
2012
|
tom 10
|
nr 3
1067-1070
EN
We prove that every Tychonoff strongly monotonically monolithic star countable space is Lindelöf, which solves a question posed by O.T. Alas et al. We also use this result to generalize a metrization theorem for strongly monotonically monolithic spaces. At the end of this paper, we study the extent of star countable spaces with k-in-countable bases, k ∈ ℤ.
4
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Remarks on absolutely star countable spaces

81%
Open Mathematics
|
2013
|
tom 11
|
nr 10
1755-1762
EN
We prove the following statements: (1) every Tychonoff linked-Lindelöf (centered-Lindelöf, star countable) space can be represented as a closed subspace in a Tychonoff pseudocompact absolutely star countable space; (2) every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented as a closed G δ-subspace in a Hausdorff (regular, Tychonoff) absolutely star countable space; (3) there exists a pseudocompact absolutely star countable Tychonoff space having a regular closed subspace which is not star countable (hence not absolutely star countable); (4) assuming $$2^{\aleph _0 } = 2^{\aleph _1 }$$, there exists an absolutely star countable normal space having a regular closed subspace which is not star countable (hence not absolutely star countable).
5
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Some weak covering properties and infinite games

81%
Open Mathematics
|
2014
|
tom 12
|
nr 2
322-329
EN
We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game Gfin(O,Do). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz spaces, Topology Appl., 1995, 64(1), 9–21], too. The selection principle S1(Do,Do) is also discussed in view of a special base. We show that every subspace of a hereditarily Lindelöf space with an ortho-base satisfies S1(Do,Do).
6
62%
Open Mathematics
|
2013
|
tom 11
|
nr 10
1750-1754
EN
We prove that the one-point Lindelöfication of a discrete space of cardinality ω 1 is homeomorphic to a subspace of C p (X) for some hereditarily Lindelöf space X if the axiom [...] holds.
7
Content available remote

LΣ(≤ ω)-spaces and spaces of continuous functions

62%
Open Mathematics
|
2010
|
tom 8
|
nr 4
754-762
EN
We present a few results and problems related to spaces of continuous functions with the topology of pointwise convergence and the classes of LΣ(≤ ω)-spaces; in particular, we prove that every Gul’ko compact space of cardinality less or equal to $$ \mathfrak{c} $$ is an LΣ(≤ ω)-space.
8
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Topological spaces compact with respect to a set of filters

52%
Open Mathematics
|
2014
|
tom 12
|
nr 7
991-999
EN
If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T 1 topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.
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