OIF spaces (Q2703087)

\textit{H. R. Bennett} and \textit{D. J. Lutzer} [Fundam. Math. 158, No.~3, 289-299 (1998; Zbl 0936.54030)] have shown that generalized ordered spaces are metrizable if and only if they have an open-in-finite (OIF) base. In the present paper the authors examine thoroughly the fundamental properties of spaces having such a base, especially their hereditary behaviour and the connection with metrizability. Among a lot of nice results and interesting examples they prove that:NEWLINENEWLINENEWLINE(1) The product of OIF spaces is an OIF space but the factors of an OIF product need not be OIF.NEWLINENEWLINENEWLINE(2) Every space \(X\) is a closed subspace of an OIF space having the same separation properties as X.NEWLINENEWLINENEWLINE(3) Every space is the open perfect image of an OIF space.NEWLINENEWLINENEWLINE(4) In \(T_2\) spaces, strong OIF bases are the same as uniform bases.NEWLINENEWLINENEWLINE(5) In \(T_3\) spaces, where all subspaces have OIF, bases, compactness, countable compactness, or local compactness will give metrizability.