On the combinatorial principle P(c)
On the compactification of closure algebras
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 complexity of subspaces of
Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space is . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of . We show that has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...
On the concept of E-regularity for fuzzy topology
On the connectedness of boundary and complement for domains
This article gives a short and elementary proof of the fact that the connectedness of the boundary of an open domain in ℝⁿ is equivalent to the connectedness of its complement.
On the connectedness of some topological spaces
On the construction of biconnected sets
On the continuity and monotonicity of restrictions of connected functions
On the continuity of the elements of the Ellis semigroup and other properties
We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of is fixed, we give a necessary and sufficient condition on a point in order that all functions of the Ellis semigroup be continuous at the given point . In the second part, we consider transitive dynamical...
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 deficiency of product spaces
On the dimension of increments of Tychonoff spaces
On the dimension of remainders in extensions of product spaces
On the existence of P-points in the Stone-Čech compactification of integers
On the existence of true uniform ultrafilters
We shall show that there is an ultrafilter on singular with countable cofinality, which cannot be reached from the set of all subuniform ultrafilters by iterating the closure of sets of size .
On the extensibility of closed filters in T spaces and the existence of well orderable filter bases
We show that the statement CCFC = “the character of a maximal free filter of closed sets in a space is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE = “closed filters in a space extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter in a space extends to a maximal closed filter with a well orderable filter base”, “for every set ,...
On the extent of separable, locally compact, selectively (a)-spaces
The author has recently shown (2014) that separable, selectively (a)-spaces cannot include closed discrete subsets of size . It follows that, assuming CH, separable selectively (a)-spaces necessarily have countable extent. However, in the same paper it is shown that the weaker hypothesis "" is not enough to ensure the countability of all closed discrete subsets of such spaces. In this paper we show that if one adds the hypothesis of local compactness, a specific effective (i.e., Borel) parametrized...
On the extent of star countable spaces
For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have...