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 To-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 homeo-morphic 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.

Additional Metadata
Journal Pacific Journal of Mathematics
Citation
Nel, L. (1972). Lattices of lower semi-continuous functions and associated topological spaces. Pacific Journal of Mathematics, 40(3), 667–673.