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.

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.

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...

The Noetherian type of topological spaces is introduced. Connections between the Noetherian type and other cardinal functions of topological spaces are obtained.

An estimate for the Novak number of a hyperspace with the Vietoris topology is given. As a consequence it is shown that this cardinal function can decrease passing from a space to its hyperspace.