Preserving continuity by Zadeh extension and invariance of separation axioms under surjections (Q695263)

With \(M\) a complete lattice, \(X\) a set, \(M^X\) the complete lattice of all maps from \(X\longrightarrow M\) under point wise ordering, each map \(f:X\longrightarrow Y\) induces two maps \(f_M^\to:M^X\longrightarrow M^Y\) and \(f_M^\gets:M^Y\longrightarrow M^X\) defined by \(f_M^\to(a)y:=\displaystyle\bigvee_{x\in f^{-1}\{y\}}a(x)\) for all \(y\in Y\) and \(f_M^\gets(b):=b\circ f\); the two maps were introduced by Zadeh with \(M=\mathbb I\) (the real unit interval) and introduction of such mappings is referred to as the Zadeh extension principle (ZEP in short). With \((L,')\) and \(\mathbb I(L)\) two complete lattices, it is shown that an \(L\)-topological space \(X\) is completely \(L\)-regular iff each lower semicontinuous function from \(X\) to \(\mathbb I(L)\) is a pointwise sup of a family of continuous functions from \(X\) to \(\mathbb I(L)\); it is further shown that complete \(L\)-regularity, \(L\)-regularity and complete \(L\)-normality are all preserved under continuous open-closed surjections for each complete lattice \((L,')\). Finally, conditions under which the Zadeh extension of a map between \(L\)-topological spaces preserves semicontinuity and continuity of functions with values in the \(L\)-unit interval are provided.