### Weak calibers and the Scott-Watson theorem

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Motivated by some examples, we introduce the concept of special almost P-space and show, using the reflection principle, that for every space $X$ of this kind the inequality “$\left|X\right|\le {\psi}_{c}{\left(X\right)}^{t\left(X\right)}$" holds.

In this paper, two cardinal inequalities for functionally Hausdorff spaces are established. A bound on the cardinality of the $\tau \theta $-closed hull of a subset of a functionally Hausdorff space is given. Moreover, the following theorem is proved: if $X$ is a functionally Hausdorff space, then $\left|X\right|\le {2}^{\chi \left(X\right)\mathit{\text{wcd}}\left(X\right)}$.

In this paper two new cardinal functions are introduced and investigated. In particular the following two theorems are proved: (i) If $\phantom{\rule{0.166667em}{0ex}}X$ 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 $X=\bigcup \{c{l}_{\tau \theta}\left(A\right):A\in {\left[S\right]}^{\le \kappa}\}$.

In this note we show the following theorem: “Let $X$ be an almost $k$-discrete space, where $k$ is a regular cardinal. Then $X$ is ${k}^{+}$-Baire iff it is a $k$-Baire space and every point-$k$ open cover $\mathcal{U}$ of $X$ such that $card\phantom{\rule{0.166667em}{0ex}}\left(\mathcal{U}\right)\le k$ is locally-$k$ at a dense set of points.” For $k={\aleph}_{0}$ we obtain a well-known characterization of Baire spaces. The case $k={\aleph}_{1}$ is also discussed.

A subset $A$ of a Hausdorff space $X$ is called an $\omega $H-set in $X$ if for every open family $\mathcal{U}$ in $X$ such that $A\subset \bigcup \mathcal{U}$ there exists a countable subfamily $\mathcal{V}$ of $\mathcal{U}$ such that $A\subset \bigcup \{\overline{V}:V\in \mathcal{V}\}$. 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 $X$.

The aim of this paper is to show, using the reflection principle, three new cardinal inequalities. These results improve some well-known bounds on the cardinality of Hausdorff spaces.

The spaces for which each $\delta $-continuous function can be extended to a $2\delta $-small point-open l.s.cṁultifunction (resp. point-closed u.s.cṁultifunction) are studied. Some sufficient conditions and counterexamples are given.

A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let $\psi $ be the following statement: “a perfect ${T}_{3}$-space $X$ with no more than ${2}^{\U0001d520}$ clopen subsets is connectifiable if and only if no proper nonempty clopen subset of $X$ is feebly compact". In this note we show that neither $\psi $ nor $\neg \psi $ is provable in ZFC.

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