Bicontinuous Function Spaces

Reinhold Heckmann, Michael Huth, and Michael Mislove


Given a sober space (XO(X)) and a complete lattice L in its Scott-topology, we study the function space [X -> L] of all continuous maps fX -> L, ordered pointwise. We show that this partial order is a bicontinuous lattice (i.e. the lattice and its order dual are continuous) if and only if L is bicontinuous, X is a continuous domain and O(X) is its Scott-topology. This extends known results on the continuity of the space [X -> L]. The techniques are novel in the theory of continuous lattices in that they employ a representation of the dual of [X -> L] as the lattice of maps preserving all suprema of type µ: O(X) -> Lop, where Lop is the order dual of L. We specialize these results down to two classes of bicontinuous lattices: linear FS-lattices and completely distributive lattices.

Key words: Continuous Lattices, Function Spaces, Completely Distributive Lattices, Linear FS-lattices

AMS classification codes: 06B35, 54C35, 06D10

[ (12p, 48k)]

Reinhold Heckmann /