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...
On the fundamental dimension of approximatively 1-connected compacta
On the level spaces of fuzzy topological spaces.
On the Noetherian type of topological spaces
The Noetherian type of topological spaces is introduced. Connections between the Noetherian type and other cardinal functions of topological spaces are obtained.
On the problem of preserving the class of continuous real functions
On the product of a compact space with an hereditarily absolutely countably compact space
We show that the product of a compact, sequential space with an hereditarily absolutely countably compact space is hereditarily absolutely countably compact, and further that the product of a compact space of countable tightness with an hereditarily absolutely countably compact -bounded space is hereditarily absolutely countably compact.
On the separation properties of
On the set-theoretic strength of the n-compactness of generalized Cantor cubes
We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product , where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set...
On the ΄΄Fragmentation΄΄ of Certain Pseydocompact Groups
On thick spaces
On topologies of free groups
On typical parametrizations of finite-dimensional compacta on the Cantor set
We prove that if X is a perfect finite-dimensional compactum, then for almost every continuous surjection of the Cantor set onto X, the set of points of maximal order is uncountable. Moreover, if X is a perfect compactum of positive finite dimension then for a typical parametrization of X on the Cantor set, the set of points of maximal order is homeomorphic to the product of the rationals and the Cantor set.