To tile consists in assembling colored tiles on
while respecting color matching. Tilings, the outcome of the tile process, can be seen as a computation model. In order to better understand the global structure of tilings, we introduce two topologies on tilings, one
Cantor and another one
Besicovitch. Our topologies are concerned with the whole set of tilings that can be generated by any tile set and are thus independent of a particular tile set. We study the properties of these two spaces and compare them. Finally, we introduce two infinite games on these spaces that are promising tools for the study of the structure of tilings.