A universal factorization theorem in topology

Beattie, M., J. E. Marsden and R. Sharpe

Canad. Math. Bull., 9, (1966), 201-207

Abstract:

The purpose of this paper is to prove and generalize the following theorem: Given any topological space X , of all T₂ spaces Z which are continuous images of X , there is a maximal one Y ; that is, one over which all others factor.

In pursuit of this result, the authors define a certain species of functors and natural transformations on the category of all topological spaces and maps. A subspecies is singled out which yields the main result. As well it leads to a uniform definition of many separation axioms, and universal proofs for some of the simple properties of these axioms.