Continuous function characterizations of stratifiable spaces (Q1872876)

Summay: The insertion of a continuous function between pairs of semi-continuous functions in a monotone manner is investigated. In particular, it is established that a topological space is stratifiable if and only if for every pair of semi-continuous functions \(\langle g,h\rangle\), \(g\) upper semi-continuous, \(h\) lower semi-continuous, and \(g\leq h\) (i.e., \(g(x)\leq h(x)\) for each \(x\)) there exists a continuous function \(\phi(\langle g,h\rangle)\) such that \(g\leq \phi(\langle g,h\rangle)\leq h\) and for each \(x\) for which \(g(x)<h(x)\) then \(g(x)<\phi(\langle g,h\rangle)(x)<h(x)\) and if \(\langle g_1,h_1\rangle\) is any other pair of semi-continuous functions with the same properties as \(g\) and \(h\) respectively and \(g\leq g_1\) and \(h\leq h_1\), then \(\phi(\langle g,h\rangle)\leq \phi(\langle g_1,h_1\rangle)\).