Change of topology, characterizations and product theorems for semilocally \(P\)-spaces.

*(English)*Zbl 0781.54007If \(P\) is a topological property, a set with this property is called a \(P\)-set. The author defines a “semilocally \(P\)-space” to be a topological space satisfying this condition: for each point \(x\) and each open neighborhood \(U\) of \(x\), there is an open neighborhood \(V\) of \(x\) which is contained in \(U\) and whose complement is the union of finitely many closed \(P\)-sets. The product theorem of the title gives conditions under which the product of such spaces is again such a space. This unifies and extends some known results for properties which include: connectedness, several forms of compactness, various versions of regularity and complete regularity.

Reviewer: P.R.Meyer (Bronx)