On countable families of sets without the Baire property

Mats Aigner ¹   ; Vitalij A. Chatyrko ¹   ; Venuste Nyagahakwa ²

¹ Department of Mathematics Linköping University 581 83 Linköping, Sweden
² Department of Mathematics National University of Rwanda Butare, Rwanda

Colloquium Mathematicum, Tome 133 (2013) no. 2, pp. 179-187

Résumé

We suggest a method of constructing decompositions of a topological space $X$ having an open subset homeomorphic to the space ($\mathbb R^n, \tau )$, where $n$ is an integer $\geq 1$ and $\tau $ is any admissible extension of the Euclidean topology of $\mathbb R^n$ (in particular, $X$ can be a finite-dimensional separable metrizable manifold), into a countable family $\mathcal F$ of sets (dense in $X$ and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of $\mathcal F$ does not have the Baire property in $X$.

Détail

DOI : 10.4064/cm133-2-4

Keywords: suggest method constructing decompositions topological space having subset homeomorphic space mathbb tau where integer geq tau admissible extension euclidean topology mathbb particular finite dimensional separable metrizable manifold countable family mathcal sets dense zero dimensional manifolds union each non empty proper subfamily mathcal does have baire property