In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If $X$ is a $T_1$ space such that (i) $L\left(X\right)t\left(X\right)\le \kappa$, (ii) $\psi \left(X\right)\le 2^{\kappa }$, and (iii) for all $A\in {\left[X\right]}^{\le 2^{\kappa }}$, $\left|\overline{A}\right|\le 2^{\kappa }$, then $\left|X\right|\le 2^{\kappa }$; and (b) (Fedeli [2]) If $X$ is a $T_2$-space then $\left|X\right|\le 2^{aql\left(X\right)t\left(X\right){\psi }_c\left(X\right)}$.

Two variations of Arhangelskii’s inequality $$\left|X\right|⩽2^{\chi \left(X\right)-L\left(X\right)}$$ 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.

We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by $d{\left(X\right)}^{\pi \chi \left(X\right)}$. This inequality improves many known results and it also solves a question by J. van Mill. We further introduce Δ-power homogeneity, which leads to a new proof of van Douwen’s theorem.

We introduce the cardinal invariant $\theta$-$a{L}^{\text{'}}\left(X\right)$, related to $\theta$-$aL\left(X\right)$, and show that if $X$ is Urysohn, then $\left|X\right|\le 2^{\theta \text{-}a{L}^{\text{'}}\left(X\right)\chi \left(X\right)}$. As $\theta$-$a{L}^{\text{'}}\left(X\right)\le aL\left(X\right)$, this represents an improvement of the Bella-Cammaroto inequality. We also introduce the classes of firmly Urysohn spaces, related to Urysohn spaces, strongly semiregular spaces, related to semiregular spaces, and weakly $H$-closed spaces, related to $H$-closed spaces.

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "$2^{ℝ}$ is countably compact" and "$2^{ℝ}$ is compact"

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...

We use the cardinal functions ac and lc, due to Fedeli, to establish bounds on the density and net weight of regular spaces which improve some well known bounds. In particular, we use the language of elementary submodels to establish that $d\left(X\right)\le \pi \chi {\left(X\right)}^{ac\left(X\right)}$ for every regular space X. This generalizes the following result due to Shapirovskiĭ: $d\left(X\right)\le \pi \chi {\left(X\right)}^{c\left(X\right)}$ for every regular space X.

We calculate the density of the hyperspace of a metric space, endowed with the Hausdorff or the locally finite topology. To this end, we introduce suitable generalizations of the notions of totally bounded and compact metric space.

We shall show that there is an ultrafilter on singular $\kappa$ with countable cofinality, which cannot be reached from the set of all subuniform ultrafilters by iterating the closure of sets of size $<\kappa$.

We show that the statement CCFC = “the character of a maximal free filter $F$ of closed sets in a $T_1$ space $(X,T)$ is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE${}_0$ = “closed filters in a $T_0$ space $(X,T)$ extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter $ℱ$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base”, “for every set $A\ne \varnothing$,...

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 "$2^{\aleph ₀}<2^{\aleph ₁}$" 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...

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...

A considerable amount of research has been done on the notions of pseudo complement, intersection and union of fuzzy sets [1], [4], [11]. Most of this work consists of generalizations or alternatives of the basic concepts introduced by L. A. Zadeh in his famous paper [13]: generalization of the unit interval to arbitrary complete and completely distributive lattices or to Boolean algebras [2]; alternatives to union and intersection using the concept of t-norms [3], [10]; alternative complements...