Subset Spaces were introduced by L. Moss and R. Parikh in . These spaces model the reasoning about knowledge of changing states.
In  a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In  the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes intersection spaces.
We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set.
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