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Register a new userOn the definability of Menger spaces which are not /sigma-compact
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Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space that is not /sigma-compact.
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A theorem by Norman L. Noble from 1970 asserts that every product of completely regular, locally pseudo-compact k_R-spaces is a k_R-space. As a consequence, all direct products of locally compact Hausdorff spaces are k_R-spaces. We provide a streamlined proof for this fact.
We study products of general topological spaces with Mengers covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
W. Hurewicz proved that analytic Menger sets of reals are $sigma$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgoz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with $omega$-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.
In our paper [18] we showed that a Tychonoff space $X$ is a $Delta$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $Delta$ of $Delta$-spaces is invariant under the basic topological operations. We prove that if $X in Delta$ and $varphi:X to Y$ is a continuous surjection such that $varphi(F)$ is an $F_{sigma}$-set in $Y$ for every closed set $F subset X$, then also $Yin Delta$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_iin Delta$, then also $Xin Delta$. In particular, $sigma$-product of any family of scattered Eberlein compact spaces is a $Delta$-space and the product of a $Delta$-space with a countable space is a $Delta$-space. Our results give answers to several open problems posed in cite{KL}. Let $T:C_p(X) longrightarrow C_p(Y)$ be a continuous linear surjection. We observe that $T$ admits an extension to a linear continuous operator $widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a $Delta$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is. Making use of obtained results, we provide a very short proof for the claim that every compact $Delta$-space has countable tightness. As a consequence, under Proper Forcing Axiom (PFA) every compact $Delta$-space is sequential. In the article we pose a dozen open questions.
We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish productive properties f
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