A space
X
is sequentially separable if there is a countable
D
⊂
X
such that every point of
X
is the limit of a sequence of points from
D
. Neither “sequential + separable” nor “sequentially ...separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.
Two variations of Arhangelskii’s inequality
for Hausdorff
X
Arhangel’skii A.V., The power of bicompacta with first axiom of countability, Dokl. Akad. Nauk SSSR, 1969, 187, 967–970 (in Russian) given ...in Stavrova D.N., Separation pseudocharacter and the cardinality of topological spaces, Topology Proc., 2000, 25(Summer), 333–343 are extended to the classes with finite Urysohn number or finite Hausdorff number.
Monotone weak Lindelöfness Bonanzinga, Maddalena; Cammaroto, Filippo; Pansera, Bruno A.
Central European journal of mathematics,
06/2011, Letnik:
9, Številka:
3
Journal Article
Recenzirano
Odprti dostop
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.
Sapirovskii 16 proved that |X|≤πχ(X)c(X)ψ(X), for a regular space X. We introduce the θ-pseudocharacter of a Urysohn space X, denoted by ψθ(X), and prove that the previous inequality holds for ...Urysohn spaces replacing the bounds on cellularity c(X)≤κ and on pseudocharacter ψ(X)≤κ with a bound on Urysohn cellularity Uc(X)≤κ (which is a weaker condition because Uc(X)≤c(X)) and on θ-pseudocharacter ψθ(X)≤κ respectively (note that in general ψ(⋅)≤ψθ(⋅) and in the class of regular spaces ψ(⋅)=ψθ(⋅)). Further, in 6 the authors generalized the Dissanayake and Willard's inequality: |X|≤2aLc(X)χ(X), for Hausdorff spaces X21, in the class of n-Hausdorff spaces and de Groot's result: |X|≤2hL(X), for Hausdorff spaces 10, in the class of T1 spaces (see Theorems 2.22 and 2.23 in 6). In this paper we restate Theorem 2.22 in 6 in the class of n-Urysohn spaces and give a variation of Theorem 2.23 in 6 using new cardinal functions, denoted by UW(X), ψwθ(X), θ-aL(X), hθ-aL(X), θ-aLc(X) and θ-aLθ(X). In 5 the authors introduced the Hausdorff point separating weight of a spaceX denoted by Hpsw(X) and proved a Hausdorff version of Charlesworth's inequality |X|≤psw(X)L(X)ψ(X)7. In this paper, we introduce the Urysohn point separating weight of a spaceX, denoted by Upsw(X), and prove that |X|≤Upsw(X)θ-aLc(X)ψ(X), for a Urysohn space X.
It is well known that separation axioms together with some local and global cardinal invariants lead to restriction of the cardinality of a given topological space. For an extensive survey, one can ...look at 16. We shall mention here just a few such results that are directly related to this paper. At the end we will discuss some long-standing open problems in cardinal invariant theory and give a possible way of approaching a solution.
Variations of selective separability Bella, Angelo; Bonanzinga, Maddalena; Matveev, Mikhail
Topology and its applications,
04/2009, Letnik:
156, Številka:
7
Journal Article
Recenzirano
Odprti dostop
A space
X is selectively separable if for every sequence
(
D
n
:
n
∈
ω
)
of dense subspaces of
X one can select finite
F
n
⊂
D
n
so that
⋃
{
F
n
:
n
∈
ω
}
is dense in
X. In this paper selective ...separability and variations of this property are considered in two special cases:
C
p
spaces and dense countable subspaces in
2
κ
.
n-H-closed spaces Basile, Fortunata Aurora; Bonanzinga, Maddalena; Carlson, Nathan ...
Topology and its applications,
03/2019, Letnik:
254
Journal Article
Recenzirano
Odprti dostop
In this paper we extend the theory of H-closed extensions of Hausdorff spaces to a class of non-Hausdorff spaces, defined in 2, called n-Hausdorff spaces. The notion of H-closed is generalized to an ...n-H-closed space. Known construction for Hausdorff spaces X, such as the Katětov H-closed extension κX, are generalized to a maximal n-H-closed extension denoted by n-κX.
Remarks on monotone (weak) Lindelöfness Bonanzinga, Maddalena; Cammaroto, Filippo; Sakai, Masami
Topology and its applications,
07/2017, Letnik:
225
Journal Article
Recenzirano
Odprti dostop
Using Erdös–Rado's theorem, we show that (1) every monotonically weakly Lindelöf space satisfies the property that every family of cardinality c+ consisting of nonempty open subsets has an ...uncountable linked subfamily; (2) every monotonically Lindelöf space has strong caliber (c+,ω1), in particular a monotonically Lindelöf space is hereditarily c-Lindelöf and hereditarily c-separable. (1) gives an answer of a question posed in Bonanzinga, Cammaroto and Pansera 3, and (2) gives partial answers of questions posed in Levy and Matveev 15. Some other properties on monotonically (weakly) Lindelöf spaces are also discussed. For example, we show that the Pixley–Roy space PR(X) of a space X is monotonically Lindelöf if and only if X is countable and every finite power of X is monotonically Lindelöf.
Answering a question of M. Sakai and partially answering a question of M. Scheepers, we use the Continuum Hypothesis (CH) to show that, for a slight modification of a space constructed by Dowker in ...1955, clopen point-cofinite covers can be diagonalized but there is a sequence of open point-cofinite covers that cannot be diagonalized.
Also this space distinguishes between Arhangelskiiʼs α2-property for Cp(X) and α2-property for Cp(X,2).
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