An answer to a question on hyperspaces
An insertion theorem for real functions
An observation on spaces with a zeroset diagonal
We say that a space has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of is countable. A space has a zeroset diagonal if there is a continuous mapping with , where . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most .
Another note on nearly paracompact spaces
Another property of the Sorgenfrey line
Another universal metacompact developable -space of weight m
Applications of limited information strategies in Menger's game
As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize -compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize -compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between -compact and Menger spaces.
Around a quotient space of Bennett-Lutzer's space.
Around Katětov's metrization theorem
Aull-paracompactness and strong star-normality of subspaces in topological spaces
We prove for a subspace of a -space , is (strictly) Aull-paracompact in and is Hausdorff in if and only if is strongly star-normal in . This result provides affirmative answers to questions of A.V. Arhangel’skii–I.Ju. Gordienko [3] and of A.V. Arhangel’skii [2].
Base-base paracompactness and subsets of the Sorgenfrey line
A topological space is called base-base paracompact (John E. Porter) if it has an open base such that every base has a locally finite subcover . It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.
Bitopological local Lindelöfness
Boolean algebras and ultracompactness
Borel extensions of Baire measures
We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom ♣ there is a Dowker space which is quasi-Mařík but not Mařík, answering a question of H. Ohta and K. Tamano, and under P(c), that there is a Mařík Dowker space, answering a question of W. Adamski. We answer further questions of H. Ohta and K. Tamano by showing that the union of a Mařík space and a compact space is Mařík,...
-compactness modulo an ideal.
Cardinal invariants of -topologies.
Cardinalities of DCCC normal spaces with a rank 2-diagonal
A topological space has a rank 2-diagonal if there exists a diagonal sequence on of rank , that is, there is a countable family of open covers of such that for each , . We say that a space satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of is countable. We mainly prove that if is a DCCC normal space with a rank 2-diagonal, then the cardinality of is at most . Moreover, we prove that if is a first countable...
Čech cohomology and covering dimension for topological spaces
Čech complete nearness spaces
We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness...
Centered-Lindelöfness versus star-Lindelöfness
We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.