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Keywords:
$p$-pseudocompactness; ultrapseudocompactness; strongly pseudocompactness; strongly $p$-pseudocompactness; weak $P$-points; $\mathfrak{c}-OK$ points; Rudin-Keisler pre-order
Summary:
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-Mascarúa A., Ultrafilters and properties related to compactness, Topology Proc. 43 (2014), 183--200] and [García-Ferreira S., Ortiz-Castillo Y.F., 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 [Discrete subsets of topological groups, Math. Notes 55 (1994), no. 1--2, 101--102].
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