On two generalizations of pseudocompactness by Salvador Garcia-Ferreira

Electronic Version 24 Summer (1999)

On two generalizations of pseudocompactness

Salvador Garcia-Ferreira

The two generalizations of pseudocompactness considered in this paper are \alpha-pseudocompactness, where \alpha is an infinite cardinal, and p-pseudocompactness, where p is an uniform ultrafilter on an infinite cardinal. Let <= _RK be the Rudin-Keisler order on \beta₍\alpha) - \alpha = \alpha^* and let P_RK(p) = { q \in \alpha^* : q <= _RK p }. We prove that if p \in U(\alpha), \alpha^{< \alpha} = \alpha, and P_RK(p) is \alpha-pseudocompact, then p is decomposable. Under GCH, if p \in U(\alpha) and P_RK(p) is \alpha-pseudocompact, then p is decomposable. It is also shown that, for every cardinal \alpha, there is p \in U(\alpha) such that P_RK(p) is \alpha-pseudocompact. Assuming a set-theoretic axiom, we may find p \in U(\omega₁) such that P_RK(p) is not \omega₁-pseudocompact. For p \in U(\omega₁), we let \beta_p(\omega) be the p-compact reflexion of the discrete space \omega. We proved that 2^\omega₁ < 2^{2^\omega} if and only if \beta_p(\omega) =/= \beta(\omega), for every p \in U(\omega₁); and CH implies that \beta_p(\omega) = \beta(\omega) for every p \in U(\omega₁). Now, for p \in U(\omega₁) and q \in \omega^*, we put \Gamma_p,q = \beta_p(\omega) - ({q} \cup \omega). These spaces satisfy the following properties:

if 2^\omega₁ < 2^{2^\omega}, then for every p \in U(\omega₁) there is q \in \omega^* such that \Gamma_p,q is p-pseudocompact;
CH implies that for every q \in \omega^*, \Gamma_p,q is not p-pseudocompact for all p \in U(\omega₁);
if q \in \omega^* is a P_\omega₂-point, then \Gamma_p,q is p-pseudocompact for every p \in U(\omega₁); and
if p \in U(\omega₁), then \Gamma_p,q is p-pseudocompact for every q \in \omega^* with \pi\chi(q, \omega^*) > omega₁.

In the last Section, we give several conditions that are equivalent to the statement ``every p-pseudocompact space is pseudocompact'', and some necessary and sufficient conditions are listed to guarantee that every p-compact space is \alpha-pseudocompact, for p \in U(\alpha).

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