Families of "shapes"

SW VandeCarr

Is there a general definition of a 'family' in topology or can a family be defined to suit one's specific requirements?

I'm thinking in reference to the Calabi Yau (CY) family of shapes which are used as a model in most string theories. CY shapes are described as a family and they have anywhere from one to over 400 holes. Given that topological genera are defined more or less by the number of holes in a manifold, what more general topological characteristic would allow for the definition of a family?