In this paper two new cardinal functions are introduced and investigated. In particular the following two theorems are proved: (i) If $X$ is a functionally Hausdorff space then $\left|X\right|\le 2^{fs\left(X\right){\psi }_{\tau }\left(X\right)}$; (ii) Let $X$ be a functionally Hausdorff space with $fs\left(X\right)\le \kappa$. Then there is a subset $S$ of $X$ such that $\left|S\right|\le 2^{\kappa }$ and $X=\bigcup \{c{l}_{\tau \theta }\left(A\right):A\in {\left[S\right]}^{\le \kappa }\}$.

It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum $X$ contains $\Delta \left(X\right)$ many pairwise disjoint dense subsets, where $\Delta \left(X\right)$ denotes the minimum size of a non-empty open set in $X$. The aim of this note is to prove the following analogous result: Every compactum $X$ contains ${\Delta }_{\delta }\left(X\right)$ many pairwise disjoint $G_{\delta }$-dense subsets, where ${\Delta }_{\delta }\left(X\right)$ denotes the minimum size of a non-empty $G_{\delta }$ set in $X$.

On the set $ℝ$ of real numbers we consider a poset ${𝒫}_{\tau }\left(ℝ\right)$ (by inclusion) of topologies $\tau \left(A\right)$, where $A\subseteq ℝ$, such that $A_1\supseteq A_2$ iff $\tau \left(A_1\right)\subseteq \tau \left(A_2\right)$. The poset has the minimal element $\tau \left(ℝ\right)$, the Euclidean topology, and the maximal element $\tau \left(\varnothing \right)$, the Sorgenfrey topology. We are interested when two topologies ${\tau }_1$ and ${\tau }_2$ (especially, for ${\tau }_2=\tau \left(\varnothing \right)$) from the poset define homeomorphic spaces $(ℝ,{\tau }_1)$ and $(ℝ,{\tau }_2)$. In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ,\tau (A\left)\right)$ is homeomorphic to the Sorgenfrey line $(ℝ,\tau (\varnothing \left)\right)$ iff $A$ is countable. We study also common...

A subset $A$ of a Hausdorff space $X$ is called an $\omega$H-set in $X$ if for every open family $𝒰$ in $X$ such that $A\subset \bigcup 𝒰$ there exists a countable subfamily $𝒱$ of $𝒰$ such that $A\subset \bigcup \{\overline{V}:V\in 𝒱\}$. In this paper we introduce a new cardinal function $t_{s\theta }$ and show that $\left|A\right|\le 2^{t_{s\theta }\left(X\right){\psi }_c\left(X\right)}$ for every $\omega$H-set $A$ of a Hausdorff space $X$.