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

Carlos Uzcátegui¹

¹ Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes Mérida 5101, Venezuela

Fundamenta Mathematicae, Tome 176 (2003) no. 1, pp. 1-16

Résumé

Let $(X,\tau )$ be a countable topological space. We say that $\tau $ is an analytic (resp. Borel) topology if $\tau $ 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_\omega $ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_\omega $. We show that $S_\omega $ 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_\omega $ has this property iff it contains a copy of $S_\omega $.

Détail

DOI : 10.4064/fm176-1-1

Keywords: tau countable topological space say tau analytic resp borel topology tau subset cantor set via characteristic functions analytic resp borel set example topology arkhangelski franklin space omega sigma delta paper study complexity sense borel hierarchy subspaces omega omega has subspaces topologies arbitrarily high borel rank has subspaces non borel topology moreover closed subset omega has property contains copy omega