Menger's basis property is a generalization of σ-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non σcompact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis.The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner.Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it selfcontained and accessible.Whenever you can settle a question by explicit construction, be not satisfied with purely existential arguments.

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A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily.We proceed to consider the class of Arhangel'skiȋ α1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let Cp(X) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our main result is that Cp(X) is an α1 space if, and only if, each Borel image of X in the Baire space is bounded. Using this characterization, we solve a variety of problems posed in the literature concerning spaces of continuous functions. 4.3. Almost continuous functions 13 4.4. wQN spaces and the Scheepers Conjecture 13 4.5. QN spaces and M spaces 14 4.6. wQN * spaces 15 4.7. Bounded-ideal convergence spaces 15 4.8. Bounded Baire-class α images 17 5. Closing the circle: Continuous bounded images again

We give topological characterizations of filters F on such that the Mathias forcing MF adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M-separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P-subspaces of w^* admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, Bonanzinga, Matveev, and Tkachuk.Comment: 7 page