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Zbl 0876.54013
Eckertson, Frederick W.; Ohta, Haruto
Weak pseudocompactness and zero sets in pseudocompact spaces. (English)
[J] Houston J. Math. 22, No.4, 737-748 (1996). ISSN 0362-1588

In this article all spaces are Tikhonov. A space $X$ is $G_\delta$-{\it dense} in a space $Y$ if $X\subset Y$ and every nonempty $G_\delta$ in $Y$ contains a point of $X$. It is a well-known result of E. Hewitt that a space $X$ is pseudocompact iff it is $G_\delta$-dense in its Stone-Čech compactification $\beta X$. A space is called weakly pseudocompact provided that it is $G_\delta$-dense in some compactification [{\it S. García-Ferreira} and {\it A. García-Máynez}, ibid. 20, No. 1, 145-159 (1994; Zbl 0809.54012)]. It is noted in the just cited paper that zero-sets in pseudocompact spaces are weakly pseudocompact, and products of weakly pseudocompact spaces are weakly pseudocompact. The authors of the paper under review study zero-sets in pseudocompact spaces and consider such questions as: \par (1) Does every weakly pseudocompact space embed as a zero-set in some pseudocompact space? \par (2) If a product is weakly pseudocompact, must one of the factors be weakly pseudocompact? \par They obtain a negative answer to (1) and present some results concerning (2). The negative answer to (1) is obtained as follows. Let $\frak b$ be the minimum cardinality of an unbounded subset of ${}^\omega\omega$ and $\frak d$ be the minimum cardinality of a $\leq^*$ cofinal subset ${}^\omega\omega$, where in each case ${}^\omega\omega$ is endowed with the relation $\leq^*$ defined by $f\leq^{*} g$ iff there is $n\in\omega$ such that $f(k)\leq g(k)$ for all $k\geq n$. The authors prove the theorem (2.1): Suppose that $Z=X\oplus\omega$ is a zero-set in $Y$, and every open cover of $X$ of cardinality at most $\frak d$ has a subcover of cardinality less than $\frak b$. Then $Y$ is not pseudocompact. Using the preceding, they show that for $\kappa=\frak d^+$, the space $\kappa\oplus\omega$ is weakly pseudocompact but cannot be embedded as a zero-set in any pseudocompact space. Next, it is shown that (2.5) for a limit ordinal $\gamma$, a space having the form $\gamma\oplus\omega$ embeds as a zero-set in some pseudocompact space iff there is a $\leq^*$-unbounded, $<^*$-increasing cf$(\gamma)$-sequence in ${}^\omega\omega$. One corollary to 2.5 is the following strengthening of a result of Peter Nyikos: If $\gamma$ is an ordinal and cf$(\gamma)=\frak b$, then $\gamma\oplus\omega$ embeds as a zero-set in a pseudocompact space. The third section focuses on cardinalities of discrete zero-sets of pseudocompact spaces. A proof is given that for each infinite cardinal $\kappa$ there is a (Mrówka-Isbell type of) pseudocompact space including a discrete zero-set $D$ with $|D|\geq\kappa$, and if $\kappa$ is an $\omega$-power, then $|D|=\kappa$. Limitations on $|D|$ are obtained: Let $\frak a$ be the minimum cardinality of an infinite mad family on $\omega$, and suppose that a pseudocompact space $X$ contains a zero-set homeomorphic to the free sum $D=\oplus\sum D$ of an infinite family of compact spaces, and $X$ contains a dense set of isolated points. Then $|\sum D|\geq\frak a$. In section four, theorems and examples are presented concerning (2) and related questions from [{\it F. W. Eckertson}, Topology Appl. 72, No. 2, 149-157 (1996; Zbl 0857.54022)]. For example, the authors prove that (4.1) every Lindelöf $G_\delta$ subset of a weakly pseudocompact space is Čech-complete, which provides a new proof that no countable power of the Sorgenfrey line is weakly pseudocompact. Theorem 4.1 is also used to show that a non-compact product of a Lindelöf and a Čech-complete Lindelöf space is not weakly pseudocompact.
[R.M.Stephenson (Columbia)]

MSC 2000:: *54C30 Real-valued functions on topological spaces
54B10 Product spaces (general topology)
54C25 Imbedding of topological spaces

Keywords: weakly pseudocompact; zero-set; Lindelöf; Čech-complete

Citations: Zbl 0809.54012; Zbl 0857.54022

Cited in: Zbl 1171.06005

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