Maddalena Bonanzinga

On the Urysohn number of a topological space

Abstract The Urysohn number of a space X is U(X) = min{τ: for every subset A ⊂ X such that |A| ≥ τ one can pick neighborhoods U a ∋ a for all a ∈ A so that Some known statements about Urysohn spaces can be generalized in terms of the Urysohn number.

Sequential + separable vs sequentially separable and another variation on selective separability

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.

On Weaker Forms of Separability

Two properties each of which is equivalent to separability for regular spaces and is weaker than separability in general are considered as well as the corresponding cardinal functions.

On the Urysohn number of a topological space II

Abstract Some variations of Arhangelskii inequality ∣X∣ = 2χ(X)L(X) for every Hausdorff space X [3], given in [2] and [6] are improved.

On a Weaker Form of Countable Compactness

A star covering property which is equivalent to countable compactness for regular spaces and weaker than countable compactness for Hausdorff spaces is introduced and considered. Various kinds of irregularity of topological spaces are discussed.

Monotone weak Lindelöfness

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.

Some remarks on generalizations of countably compact spaces and Lindelöf spaces

We prove some propositions and present some examples concerning the properties between countably compact and pseudocompact and the properties between Lindelöf and pseudo-Lindelöf.

Combinatorics of Thin Posets: Application to Monotone Covering Properties

A class of posets, called thin posets, is introduced, and it is shown that every thin poset can be covered by a finite family of trees. This fact is used to show that (within ZFC) every separable monotonically Menger space is first countable. This contrasts with the previously known fact that under CH there are countable monotonically Lindelöf spaces which are not first countable.

Chapter 2 – Problems on star-covering properties

This chapter discusses some questions about properties defined in terms of stars with respect to open covers. If u is a cover of X , and A a subset of X , then St(A , u ) = St 1 ( A , u ) = U{ U ∈ u : U ∩ A ≠ O ;}. For n = 1 , 2 , . . . , St n + 1 ( A , u ) = St(St n ( A , u), u). Even if many properties can be characterized in terms of stars and normality is equivalent to the requirement that every finite open cover has an open star refinement, the discussion focuses mostly on properties specifically defined by means of stars. All spaces are assumed to be Tychonoff unless a weaker axiom of separation is indicated. If A is an almost disjoint family of infinite subsets of ω, then Ψ ( A ) denotes the associated Ψ‑space. The chapter describes in detail the concepts related to compactness-type properties. It details about Lindelof-type properties and cardinal functions. Concepts of paracompactness-type properties are also explained in the chapter.

On the cardinality of n-Urysohn and n-Hausdorff spaces

Two variations of Arhangelskii’s inequality