In this paper the lattice of all
real-valued lower semi-continuous functions on a topological space is studied. It is
first shown that there is no essential loss if attention is restricted to T0-spaces. By
suitably topologizing a certain set of equivalence classes of prime ideals, it is shown
that a topological space is determined by the lattice. This topological space is
homeomorphic with the original space X whenever X has the property that every
non-empty irreducible closed set is a point closure. The sublattices of functions
taking values only in intervals of the form (a,b] and [a,b] are compared. Relations
between the above function lattices and the lattice of all closed subsets are also
discussed.
