# Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 2, page 353-372
- ISSN: 1435-9855

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topTsaban, Boaz, and Zdomsky, Lubomyr. "Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations." Journal of the European Mathematical Society 014.2 (2012): 353-372. <http://eudml.org/doc/277316>.

@article{Tsaban2012,

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 $\sigma _1$ 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.},

author = {Tsaban, Boaz, Zdomsky, Lubomyr},

journal = {Journal of the European Mathematical Society},

keywords = {pointwise convergence; point-cofinite covers; $\alpha _1$; eventual dominance; Hurewicz property; selection principles; QN sets; ideal convergence; pointwise convergence; point-cofinite covers; space; eventual dominance; Hurewicz property; selection principles; QN sets; ideal convergence},

language = {eng},

number = {2},

pages = {353-372},

publisher = {European Mathematical Society Publishing House},

title = {Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations},

url = {http://eudml.org/doc/277316},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Tsaban, Boaz

AU - Zdomsky, Lubomyr

TI - Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 2

SP - 353

EP - 372

AB - 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 $\sigma _1$ 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.

LA - eng

KW - pointwise convergence; point-cofinite covers; $\alpha _1$; eventual dominance; Hurewicz property; selection principles; QN sets; ideal convergence; pointwise convergence; point-cofinite covers; space; eventual dominance; Hurewicz property; selection principles; QN sets; ideal convergence

UR - http://eudml.org/doc/277316

ER -

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