The axioms in Kelley's General Topology

Statement

The axioms and definitions in this section are, but for a few inessential details, taken from the Appendix to Kelley (1955). The explanatory remarks below are not his. The Appendix states 181 theorems and definitions, and warrants careful reading as an abbreviated exposition of axiomatic set theory by a working mathematician of the first rank. Kelley introduced his axioms gradually, as needed to develop the topics listed after each instance of Develop below.

Notations appearing below and now well-known are not defined. Peculiarities of Kelley's notation include:

• He did not distinguish variables ranging over classes from those ranging over sets;
• domain f and range f denote the domain and range of the function f; this peculiarity has been carefully respected below;
• His primitive logical language includes class abstracts of the form $\ \{x : A(x)\},$ "the class of all sets x satisfying A(x)."

If the scope of all quantified variables in the above axioms is restricted to sets, all axioms except III and the schema IV are ZFC axioms. IV is provable in ZFC. Hence the Kelley treatment of MK makes very clear that all that distinguishes MK from ZFC are variables ranging over proper classes as well as sets, and the Classification schema.