On the complexity of subspaces of $S_{\omega }$

Carlos Uzcátegui. "On the complexity of subspaces of $S_{ω}$." Fundamenta Mathematicae 176.1 (2003): 1-16. <http://eudml.org/doc/282681>.

@article{CarlosUzcátegui2003,
abstract = {Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_ω$ is $F_\{σδ\}$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_ω$. We show that $S_ω$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover, a closed subset of $S_ω$ has this property iff it contains a copy of $S_ω$.},
author = {Carlos Uzcátegui},
journal = {Fundamenta Mathematicae},
keywords = {countable topological space; sequential space; Borel set; analytic set},
language = {eng},
number = {1},
pages = {1-16},
title = {On the complexity of subspaces of $S_\{ω\}$},
url = {http://eudml.org/doc/282681},
volume = {176},
year = {2003},
}

TY - JOUR
AU - Carlos Uzcátegui
TI - On the complexity of subspaces of $S_{ω}$
JO - Fundamenta Mathematicae
PY - 2003
VL - 176
IS - 1
SP - 1
EP - 16
AB - Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_ω$ is $F_{σδ}$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_ω$. We show that $S_ω$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover, a closed subset of $S_ω$ has this property iff it contains a copy of $S_ω$.
LA - eng
KW - countable topological space; sequential space; Borel set; analytic set
UR - http://eudml.org/doc/282681
ER -