Compliments to bad spaces

Could mathematics be made any better by a different choice of basic definitions? Definitions in some mathematical theories exclude any mentioning of objects, which are believed to be nasty. We will consider few examples of such "political correctness." Speaking on differential manifolds, we usually pretend that they have no singular siblings. This causes lots of inconveniences. Another example is finite topological spaces. Most of mathematicians believe that all finite topological spaces are either trivial or nasty. Topology appears to be the only mathematical field that feels ashamed of its finite objects.