It is proved a rather general version of the statement that if the union of arbitrary elements of a system λ always belongs
to λ then the intersections of elements of λ constitute an ultratopology (i.e. a topology where intersections of open sets
Let a topological s ace X be said to be rarophile i .each rare set is a .nite intersection of semi-o en sets (in the sense that A is semi-o en i .A .cl(int(A))).Various characteri- zations for raro hile spaces,examples of rarophile and non-rarophile spaces,ro erties of raro hile spaces are given and some o en roblems formulated.
The author examines sets with the property int A int A, where denotes a map- ping with suitable properties from exp X to exp X in a topological space X. Thespecial case =cl is ofparticular importance.