On pseudocompact, countably compact, locally bicompact mappings, and -mappings.
On pseudocompactness and related notions in ZF
We study in ZF and in the class of spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.
On pushing out frames
On R-compact spaces
On regular and sigma-smooth two valued measures and lattice generated topologies.
On -cluster sets and -closed spaces.
On semicompact sets and associated properties.
On sets of confluent and related mappings in the space
On shapes of topological spaces
On some classes of compact spaces lying in -products
On some modifications of fuzzy topology
On some notions related to compactness for locales
On some subspaces of the Helly space
On structure space of -semigroups
In this paper we introduce the notion of the structure space of -semigroups formed by the class of uniformly strongly prime ideals. We also study separation axioms and compactness property in this structure space.
On sub-, pseudo- and quasimaximal spaces
The structure of sub-, pseudo- and quasimaximal spaces is investigated. A method of constructing non-trivial quasimaximal spaces is presented.
On subcompactness and countable subcompactness of metrizable spaces in ZF
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space is countably compact if and only if it is countably subcompact relative to . (iii) For every metrizable space , the following are equivalent: (a) is compact; (b) for every open filter of , ; (c) is subcompact relative to . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...
On the cardinality and weight spectra of compact spaces, II
Let B(κ,λ) be the subalgebra of P(κ) generated by . It is shown that if B is any homomorphic image of B(κ,λ) then either or ; moreover, if X is the Stone space of B then either or . This implies the existence of 0-dimensional compact spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.
On the class of continuous images of Valdivia compacta.
On the Compactness and Countable Compactness of in ZF
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " is countably compact" and " is compact"
On the convergence and character spectra of compact spaces
An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our results (X...