Vol. 40, No. 3, 1972

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Lattices of lower semi-continuous functions and associated topological spaces

Louis D. Nel

Vol. 40 (1972), No. 3, 667–673

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.

Mathematical Subject Classification 2000
Primary: 54C40
Secondary: 06A20
Received: 11 February 1971
Published: 1 March 1972
Louis D. Nel