Hereditarily Hurewicz spaces and Arhangel'skiĭ sheaf amalgamations

Abstract

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

On Ohta–Sakai's properties of a topological space

“…Moreover, by A. Dow [15] Scheepers' Conjecture holds in the Laver model constructed by A. Laver [24], see e.g. [28,44] or [4].…”

“…We summarize essential results about these properties, for more see [18,6,34,3,35,44]. For a definition of an S 1 (Γ, Γ)-space see the end of this section.…”

“…Actually, by [8] and [44] we have L. Bukovský, I. Recław and M. Repický [7] showed that any perfectly normal wQN-space has Hurewicz property. Thus the Hurewicz property in Theorem 1.3 can be replaced by wQN-space.…”

The Ramsey property for C p (X)

“…Theorem 1.4 [22]. For a perfectly normal and strongly zero-dimensional space X, the following are equivalent.…”

“…Gerlits, Nagy [11] and Scheepers [19] showed that properties (α 2 ) and (α 4 ) are equivalent for C p (X). Tsaban and Zdomskyy [22] showed that if X is perfectly normal and strongly zero-dimensional (i.e., X is a space satisfying that every open subset is the union of countably many clopen sets in X), then properties (α 1 ) and (α 3/2 ) are equivalent for C p (X). Indeed, they proved the following.…”