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 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.
Cite this article
Boaz Tsaban, Lyubomyr Zdomskyy, Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 353–372DOI 10.4171/JEMS/305