Given universal set $U$. Is there any name of the collection of subsets of $U$ (call them quasi-open) satisfying the following axioms:


i) $\emptyset$ and $U$ are quasi-open;


ii) finite intersections of quasi-open sets are quasi-open;


iii) countable unions of quasi-open sets are quasi-open?



"countable" topology
This type of structure has sometimes been called a "$\sigma$-topology", especially in connection with ultraproducts. For instance the invariant Loeb measure on an ultrafinite group can be thought of as a sort of Haar measure on a $\sigma$-topological group. See for instance Section 2 of this paper of Bergelson and Tao:


<http://arxiv.org/abs/1211.6372>


