## Dynamic topological S5

The topological semantics for modal logic interprets a standard modal propositional
language in topological spaces rather than Kripke frames: the most general logic of
topological spaces becomes S4. But other modal logics can be given a topological semantics
by restricting attention to subclasses of topological spaces: in particular, S5 is logic
of the class of *almost discrete* topological spaces, and also of *trivial* topological spaces.
Dynamic Topological Logic (DTL) interprets a modal language enriched with two unary
temporal connectives, *next* and *henceforth*. DTL interprets the extended language in
*dynamic topological systems*: a DTS is a topological space together with a continuous
function used to interpret the temporal connectives. In this paper, we axiomatize four
conservative extensions of S5, and show them to be the logic of continuous functions
on almost discrete spaces, of homeomorphisms on almost discrete spaces, of continuous
functions on trivial spaces and of homeomorphisms on trivial spaces.