also has enough projectives. This is true since, for every left (right) -module , we can take to be the free (and hence projective) -module generated by a generating set for (for example we can take to be ). Then the canonical projection is the required surjection. The projective objects in the category of compact Hausdorff spaces are precisely the extremally disconnected spaces. This result is due to , with a simplified proof given by . In the category of Banach spaces and contractions (i.e., functionals whose norm is at most 1), the epimorphisms are precisely the maps with dense image. shows that the zero space is the only projective object in this category. There are non-trivial spaces, though, which are projective with respect to the class of surjective contractions. In the category of normed vector spaces with contractions (and surjective maps as "surjections"), the projective objects are precisely the -spaces. References External links ' Homological algebra Objects (category theory)