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'skiĭ ${\alpha }_1$ 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 ${\alpha }_1$ 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.