Semi-topological properties (Q1187586)

Summary: A property preserved under a semi-homeomorphism is said to be a semi- topological property. We prove the following results: (1) A topological property \(P\) is semi-topological if and only if the statement ``\((X,{\mathcal T})\) has \(P\) if and only if \((X,F({\mathcal T}))\) has \(P\)'' is true where \(F({\mathcal T})\) is the finest topology on \(X\) having the same family of semi-open sets as \((X,{\mathcal T})\). (2) If \(P\) is a topological property being minimal \(P\) is semi-topological if and only if for each minimal \(P\) space \((X,{\mathcal T})\), \({\mathcal T}=F({\mathcal T})\).