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Register a new userSmall Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $beta omega$
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We study the relations between a generalization of pseudocompactness, named $(kappa, M)$-pseudocompactness, the countably compactness of subspaces of $beta omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $mathfrak c$-many selective ultrafilters, that there exists a subspace of $beta omega$ that is $(kappa, omega^*)$-pseudocompact for all $kappa<mathfrak c$, but $text{CL}(X)$ isnt pseudocompact. We prove in ZFC that if $omegasubseteq Xsubseteq betaomega$ is such that $X$ is $(mathfrak c, omega^*)$-pseudocompact, then $text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $mathfrak c$ for some small cardinals. We provide an example of a subspace of $beta omega$ for which all powers below $mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $omega subseteq X$, the pseudocompactness of $text{CL}(X)$ implies that $X$ is $(kappa, omega^*)$-pseudocompact for all $kappa<mathfrak h$, and provide an example of such an $X$ that is not $(mathfrak b, omega^*)$-pseudocompact.
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