Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

TL;DR

This paper characterizes hereditarily Hurewicz spaces via bounded Borel images in the Baire space and explores Arhangel'skii alpha_1 spaces, linking function space properties to boundedness conditions.

Contribution

It provides a new characterization of hereditarily Hurewicz spaces and establishes a criterion for C_p(X) being alpha_1 based on Borel images, solving existing open problems.

Findings

Hereditarily Hurewicz spaces characterized by bounded Borel images.

C_p(X) is alpha_1 if and only if Borel images of X are bounded.

Resolved several open questions about spaces of continuous functions.

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 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.