# Hereditarily Hurewicz spaces and Arhangel'skiĭ sheaf amalgamations

### Boaz Tsaban

Weizmann Institute of Science, Rehovot, Israel### Lyubomyr Zdomskyy

Weizmann Institute of Science, Rehovot, Israel

## 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 $α_{1}$ 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 $α_{1}$ 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'skiĭ sheaf amalgamations. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 353–372

DOI 10.4171/JEMS/305