  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 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.
