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On H-closed and minimal Hausdorff spaces and the Boolean prime ideal theorem. (English) Zbl 1454.03063

In what follows, ZF is Zermelo-Fraenkel set theory, AC is the axiom of choice and ZFC = ZF + AC. BPI is the Boolean prime ideal theorem, which states that every non-trivial Boolean algebra has a prime ideal. BPI is equivalent to Tarski’s ultrafilter theorem, which asserts that “every filter on any set can be extended to an ultrafilter”. \(\mathrm{AC}_{\mathrm{fin}}\) is the restriction of the axiom of choice to families of non-empty finite sets.
All topological spaces are assumed to be Hausdorff. An extension of a topological space \(X\) is a space \(Y\) which contains (up to homeomorphism) \(X\) as a dense subspace. \(X\) is H-closed if it is closed in very Hausdorff space in which \(X\) can be embedded. It was shown by P. Alexandroff and P. Urysohn [Math. Ann. 92, 258–266 (1924; JFM 50.0128.06)] that a Hausdorff space is H-closed if, and only if, every open cover has a finite subfamily with dense union (and therefore a H-closed space is compact if, and only if, it is regular). A Hausdorff space \((X,\tau)\) is said to be minimal if there are no Hausdorff topologies on \(X\) which are strictly coarser than \(\tau\), or, equivalently, \((X,\tau)\) is minimal if \(\tau\) is a minimal element of the lattice of all Hausdorff topologies on \(X\) partially ordered by \(\subseteq\). It is well-known that compact Hausdorff spaces are minimal. M. Katětov has shown in [Čas. Mat. Fys. 69, 36–49 (1940; Zbl 0022.41203)] that minimal Hausdorff spaces are H-closed and that a Hausdorff space is minimal if, and only if, it is H-closed and semiregular (i.e., the family of all regular open sets is a base for \(\tau\)).
If \((X,\tau)\) is a topological space, a non-empty family of non-empty open sets \(\mathcal{F}\) is said to be an open filter on \(X\) if it is a filter in the lattice of open sets of \(X\) partially ordered by \(\subseteq\). If \(\mathcal{F}\) is an open filter on \(X\), the adherence of \(\mathcal{F}\), denoted by \(a(\mathcal{F})\), is given by \(\bigcap\{\overline{O}: O \in \mathcal{F}\}\), and any element of \(a(\mathcal{F})\) is said to be an adherent point of \(\mathcal{F}\). If \(a(\mathcal{F}) = \emptyset\), the open filter \(\mathcal{F}\) is said to be free. If \(x \in X\) and every open neighbourhood of \(x\) belongs to the open filter \(\mathcal{F}\) then we say that \(\mathcal{F}\) converges to \(x\).
An open filter \(\mathcal{F}\) on \(X\) is said to be an open ultrafilter on \(X\) if it is a maximal element in the family of all open filters on \(X\) partially ordered by \(\subseteq\). It was (fair recently) shown by Y. T. Rhineghost [Cah. Topol. Géom. Différ. Catég. 43, No. 4, 313–315 (2002; Zbl 1029.03037)] that, in ZF, BPI is equivalent to each one of “For each non-empty topological space, its lattice of open sets contans an ultrafilter” and “For topological spaces, each open filter can be extended to an open ultrafilter”. It is known that, within ZFC, a Hausdorff space is H-closed if, and only if, every open ultrafilter on the space converges [N. Bourbaki, Éléments de mathématique. Fasc. II. Premiere partie. Livre III: Topologie générale. 3ieme ed. entièrement refondue. Paris: Hermann & Cie (1961; Zbl 0102.37603)].
Let \(X\) be a space. Two extensions \(Y_1\) and \(Y_2\) are called equivalent if there is a homeomorphism \(f:Y_1 \to Y_2\) such that \(f \upharpoonright X = \text{id}_X\) (the identity function on \(X\)). If \(Y\) and \(Z\) are two extensions of \(X\), \(Y\) is said to be projectively larger then \(Z\) (denoted by \(Z \leq Y\)) if there is a continuous function \(f:Y \to Z\) such that \(f\upharpoonright X = \mathrm{id}_X\). The binary relation \(\leq\) is a partial order on the quotient class of extensions of \(X\).
Let \(X\) be an space. Let \(kX\) be the disjoint union \[ kX = X \cup \{\mathcal{F}: \mathcal{F}\text{ is a free open ultrafilter on }X\} \] and let \(\mathcal{Q}_X\) be the topology on \(kX\) generated by the base \[ \mathcal{B}_X = \tau \cup \{\{\mathcal{F}\}\cup W: W \in \mathcal{F}\text{ and }\mathcal{F} \in kX \setminus X\}. \]
The space \((kX,\mathcal{Q}_X)\) is the Katětov extension of \(X\).
In the paper under review, the author investigates whether BPI is equivalent to a number of fundamental results in the area of H-closed and minimal Hausdorff spaces, and he declares that, to the best of his knowledge, there is no published work in the literature which discusses the set-theoretic strength of results of this area in comparison with AC or weak forms of AC. He also notes that there is no mention of results related to H-closed and minimal Hausdorff spaces in the list of forms of the standard reference [P. Howard and J. E. Rubin, Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)].
The main results of the paper are theorems stating that, in ZF, each one of the following statements is equivalent to BPI:
(i)
A Hausdorff space is H-closed if, and only if, every open ultrafilter on the space converges;
(ii)
Products of H-closed Hausdorff spaces are H-closed;
(iii)
Products of minimal Hausdorff spaces are minimal;
(iv)
For every Hausdorff space \(X\), the Katětov space \(kX\) is an H-closed extension of \(X\); and
(v)
Every Hausdorff space has a (unique up to homeomorphism) projectively largest Katětov H-closed extension.

The following implications are also established in the paper: BPI \(\Rightarrow\) “products of non-empty \(H\)-closed spaces are non-empty” \(\Rightarrow\) “products of non-empty minimal spaces are non-empty” \(\Rightarrow\) \(\mathrm{AC}_{\mathrm{fin}}\).

MSC:

03E25 Axiom of choice and related propositions
54B10 Product spaces in general topology
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D30 Compactness
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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References:

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