Lattices of lower semi-continuous functions and associated topological spaces

Louis Nel
1972 Pacific Journal of Mathematics  
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 homeomorphic with the original space X whenever X has the property that every non-empty irreducible closed set is a point
more » ... d 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.
doi:10.2140/pjm.1972.40.667 fatcat:j5lju26hhvfh5pd4w7es2hrwke