Abstract:For a free ultrafilter $p$ on $\mathbb{N}$, the concepts of strong pseudocompactness, strong $p$-pseudocompactness and pseudo-$\omega$-boundedness were introduced in [Angoa J., Ortiz-Castillo Y.F., Tamariz-Mascar\'ua A., {\it Ultrafilters and properties related to compactness\/}, Topology Proc. {\bf 43} (2014), 183--200] and [Garc\'\i a-Ferreira S., Ortiz-Castillo Y.F., {\it Strong pseudocompact properties of certain subspaces of $\mathbb N^*$}, submitted]. These properties in a space $X$ characterize the pseudocompactness of the hyperspace $\mathcal{K}(X)$ of compact subsets of $X$ with the Vietoris topology. In this paper, we study the strong pseudocompactness and strong $p$-pseudocompactness of certain spaces. Besides, we established a relationship between these kind of properties and a result involving topological groups of I.~Protasov [{\it Discrete subsets of topological groups\/}, Math. Notes {\bf 55} (1994), no.~1--2, 101--102].
Keywords: $p$-pseudocompactness; ultrapseudocompactness; strongly pseudocompactness; strongly $p$-pseudocompactness; weak $P$-points; $\mathfrak{c}-OK$ points; Rudin-Keisler pre-order
AMS Subject Classification: 54A20 54D99 54D80