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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.

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 $𝒰$ of $X$ such that $card\left(𝒰\right)\le k$ 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.

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 $X=\bigcup \{c{l}_{\tau \theta }\left(A\right):A\in {\left[S\right]}^{\le \kappa }\}$.

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 $\left|X\right|\le 2^{\chi \left(X\right)\mathit{\text{wcd}}\left(X\right)}$.

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.

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 $X$.

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 $2^{𝔠}$ 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.