On the problem of compact totally disconnected reflection of nonmetrizability. (English) Zbl 1354.54021
We assume the reader is familiar with the consistent (meaning, relatively consistent with \(\mathbf{ZFC}\)) statement \(\mathbf{MA + \neg CH}\) – that is, the conjunction of Martin’s Axiom and the negation of the Continuum Hypothesis. A Luzin set is an uncountable subset of the reals which intersects every meager set on a countable subset. The existence of Luzin sets follows from \(\mathbf{CH}\) and it is inconsistent with \(\mathbf{MA + \neg CH}\). The Open Colouring Axiom (denoted by \(\mathbf{OCA}\)) is the following statement: For any partition of the family of unordered pairs of a subset \(X\) of the reals into two subsets \(K_0\) and \(K_1\) (that is, \(X \subseteq \mathbb{R}\) and \([X]^2 = K_0 \cup K_1\), with \(K_0 \cap K_1 = \emptyset\)) such that \(K_0\) is open in the product topology on \(X \times X\) there is either an uncountable \(Y \subseteq X\) such that \([Y]^2 \subseteq K_0\) or \(X = \bigcup\limits_{n \in \mathbb{N}} X_n\) where \([X_n]^2 \subseteq K_1\) for each \(n \in \mathbb{N}\). \(\mathbf{OCA}\) is consistent with \(\mathbf{MA}\), implies the failure of \(\mathbf{CH}\) and follows from the Proper Forcing Axiom (\(\mathbf{PFA}\)) or Martin’s Maximum (\(\mathbf{MM}\)).
The paper under review investigates reflection phenomena on non-metrizable spaces. As it is usual in the context of reflection results, the general question is: must a non-metrizable space have some nice non-metrizable subspace? For instance, it was shown by A. Dow [Proc. Am. Math. Soc. 104, No. 3, 999–1001 (1988; Zbl 0692.54018)] that every non-metrizable compact Hausdorff space has, necessarily, a non-metrizable subspace of cardinality \(\aleph_1\). In the paper, it is asked about reflection of non-metrizability for compact Hausdorff spaces to totally disconnected subspaces or quotients; more specifically, the main questions are the following:
Question. Suppose that \(K\) is a compact Hausdorff space which is non-metrizable.
\((1)\) Is there \(L \subseteq K\) which is compact, non-metrizable and totally disconnected?
\((2)\) Is there a closed subspace \(K' \subseteq K\) and a continuous surjective map \(\phi: K' \to L\) such that \(L\) is non-metrizable and totally disconnected?
A number of results relating (2) of the above Question to the structure of the Banach space \(C(K)\) are established in the paper under review. The author emphasizes that his paper can be summarized as an attempt to construct spaces providing negative answers to \((1)\) and \((2)\) of the above question, and under this point of view the main results are, as pointed out by the author himself, the following:
\((a)\) There is (in \(\mathbf{ZFC}\)) a non-metrizable compact Hausdorff space where every totally disconnected compact subspace is metrizable. The given example is an unordered split interval.
\((b)\) Assuming the existence of a Luzin set there is an unordered split interval which is a non-metrizable compact Hausdorff space without a continuous image containing a nonmetrizable totally disconnected closed subspace.
\((c)\) It is consistent with \(\mathbf{MA + \neg CH}\) that there is a nonmetrizable compact space with no non-metrizable totally disconnected subspace in any of its continuous images.
\((d)\) Assuming \(\mathbf{OCA}\) every non-metrizable split compact space has a continuous image with a non-metrizable totally disconnected closed subspace.
The paper under review investigates reflection phenomena on non-metrizable spaces. As it is usual in the context of reflection results, the general question is: must a non-metrizable space have some nice non-metrizable subspace? For instance, it was shown by A. Dow [Proc. Am. Math. Soc. 104, No. 3, 999–1001 (1988; Zbl 0692.54018)] that every non-metrizable compact Hausdorff space has, necessarily, a non-metrizable subspace of cardinality \(\aleph_1\). In the paper, it is asked about reflection of non-metrizability for compact Hausdorff spaces to totally disconnected subspaces or quotients; more specifically, the main questions are the following:
Question. Suppose that \(K\) is a compact Hausdorff space which is non-metrizable.
\((1)\) Is there \(L \subseteq K\) which is compact, non-metrizable and totally disconnected?
\((2)\) Is there a closed subspace \(K' \subseteq K\) and a continuous surjective map \(\phi: K' \to L\) such that \(L\) is non-metrizable and totally disconnected?
A number of results relating (2) of the above Question to the structure of the Banach space \(C(K)\) are established in the paper under review. The author emphasizes that his paper can be summarized as an attempt to construct spaces providing negative answers to \((1)\) and \((2)\) of the above question, and under this point of view the main results are, as pointed out by the author himself, the following:
\((a)\) There is (in \(\mathbf{ZFC}\)) a non-metrizable compact Hausdorff space where every totally disconnected compact subspace is metrizable. The given example is an unordered split interval.
\((b)\) Assuming the existence of a Luzin set there is an unordered split interval which is a non-metrizable compact Hausdorff space without a continuous image containing a nonmetrizable totally disconnected closed subspace.
\((c)\) It is consistent with \(\mathbf{MA + \neg CH}\) that there is a nonmetrizable compact space with no non-metrizable totally disconnected subspace in any of its continuous images.
\((d)\) Assuming \(\mathbf{OCA}\) every non-metrizable split compact space has a continuous image with a non-metrizable totally disconnected closed subspace.
Reviewer: Samuel Gomes da Silva (Salvador)
MSC:
| 54D30 | Compactness |
| 03E35 | Consistency and independence results |
| 46B25 | Classical Banach spaces in the general theory |
Keywords:
totally disconnected; reflection; split interval; Banach spaces of continuous functions; Luzin set; open coloring axiomCitations:
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