  The thesis presents the subject of synthetic topology, especially with
relation to metric spaces. A model of synthetic topology is a categorical model
in which objects possess an intrinsic topology in a suitable sense, and all
morphisms are continuous with regard to it. We redefine synthetic topology in
order to incorporate closed sets, and several generalizations are made. Real
numbers are reconstructed (to suit the new background) as open Dedekind cuts.
An extensive theory is developed when metric and intrinsic topology match. In
the end the results are examined in four specific models.
