The notion of quasi-p-boundedness for p ∈ ${\omega }^{*}$ is introduced and investigated. We characterize quasi-p-pseudocompact subsets of β(ω) containing ω, and we show that the concepts of RK-compatible ultrafilter and P-point in ${\omega }^{*}$ can be defined in terms of quasi-p-pseudocompactness. For p ∈ ${\omega }^{*}$, we prove that a subset B of a space X is quasi-p-bounded in X if and only if B × $P_{RK}\left(p\right)$ is bounded in X × $P_{RK}\left(p\right)$, if and only if $c{l}_{\beta \left(X\times P_{RK}\left(p\right)\right)}\left(B\times P_{RK}\left(p\right)\right)=c{l}_{\beta X}B\times \beta \left(\omega \right)$, where $P_{RK}\left(p\right)$ is the set of Rudin-Keisler predecessors of p.

Properties similar to countable fan-tightness are introduced and compared to countable tightness and countable fan-tightness. These properties are also investigated with respect to function spaces and certain classes of continuous mappings.

A sufficient condition that the product of two compact spaces has the property of weak approximation by points (briefly WAP) is given. It follows that the product of the unit interval with a compact WAP space is also a WAP space.

Let B(κ,λ) be the subalgebra of P(κ) generated by ${\left[\kappa \right]}^{\le \lambda }$. It is shown that if B is any homomorphic image of B(κ,λ) then either $\left|B\right|<2^{\lambda }$ or $\left|B\right|={\left|B\right|}^{\lambda }$; moreover, if X is the Stone space of B then either $\left|X\right|\le 2^{2^{\lambda }}$ or $\left|X\right|=\left|B\right|={\left|B\right|}^{\lambda }$. This implies the existence of 0-dimensional compact $T_2$ spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.

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

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.