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'skii $\alpha\_1$ spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C\_\_p(X) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our main result is that C\_\_p(X) is an \_σ\_1 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.