 
Summary: Spaces to Play: Topogames
Marco Aiello 1 and Johan van Benthem
ILLC, University of Amsterdam
Logical games may provide a useful paradigm to analyzeand also enjoy
topology. We have been investigating EhrenfeuchtFra¨iss'e style model compari
son games for topological models (topological spaces equipped with a valuation
function) of modal languages.
1. TopoGames: the rules
Spoiler and Duplicator play over two topological models hX; O; ši; hX 0 ; O 0 ; š 0
i
starting from two given points x 2 X; x 0
2 X 0 , which we call current points,
for a given number of rounds n. We refer to such game as TG(X;X 0 ; n; x; x 0 ).
Intuitively, Spoiler is trying to prove that the two points are `topologically'
different, while Duplicator is doing the opposite. Spoiler starts by choosing a
model containing the current point in that model. Duplicator replies by an
open set in the other space also containing the current point. The round is
not over yet, as Spoiler has now to pick a point within Duplicator's open. The
new current point of that model. Duplicator replies by picking a corresponding
point in Spoiler's open. The new current point of that model. The first round
