Bruno A. Pansera

Weaker forms of the Menger property

Abstract In this paper we study a weaker form of the classical concept of Menger property. This property, called weakly Menger, is independent from the Menger property and the almost Menger property. In particular, for Tychonoff spaces weaker Menger spaces are not equivalent to almost Menger spaces. We give some characterizations in terms of regular open sets and almost continuous mappings. We also introduce a weaker form of the star-Menger property and the notion of almost γk -set. Some open questions are given.

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.

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.

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

Two variations of Arhangelskii’s inequality

Weak Covering Properties and Selection Principles

\left| X \right| \leqslant 2^{\chi (X) - L(X)}

Monotone partitions and almost partitions

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.