## 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'skiǐ α1 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 α1 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.

Original language | English |
---|---|

Pages (from-to) | 353-372 |

Number of pages | 20 |

Journal | Journal of the European Mathematical Society |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - 2012 |

## Keywords

- Eventual dominance
- Hurewicz property
- Ideal convergence
- Point-cofinite covers
- Pointwise convergence
- QN sets
- Selection principles
- α 1