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.

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