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Keywords:
$p$-pseudocompactness; ultrapseudocompactness; strongly pseudocompactness; strongly $p$-pseudocompactness; weak $P$-points; $\mathfrak{c}-OK$ points; Rudin-Keisler pre-order
$p$-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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