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Keywords:
completely distributive lattice; continuous function; continuous lattice; Scott topology; subset system; $\mathcal Z$-continuous; $\mathcal Z$-distributive
completely distributive lattice; continuous function; continuous lattice; Scott topology; subset system; $\mathcal Z$-continuous; $\mathcal Z$-distributive
Summary:
It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal O X$ are continuous lattices. This result extends to certain classes of $\mathcal Z$-distributive lattices, where $\mathcal Z$ is a subset system replacing the system $\mathcal D$ of all directed subsets (for which the $\mathcal D$-distributive complete lattices are just the continuous ones). In particular, it is shown that if $[X,Y]$ is a complete lattice then it is supercontinuous (i.e.\^^Mcompletely distributive) iff both $Y$ and $\mathcal O X$ are supercontinuous. Moreover, the Scott topology on $Y$ is the only one making that equivalence true for all spaces $X$ with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for $[X,Y]$ to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations.
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