An elementary theory of the category of topological spaces
Dana I. Schlomiuk · Transactions of the American Mathematical Society · 1970
An elementary system of axioms was given by F. W. Lawvere for the category of sets and mappings. The purpose of this paper is to provide a finite number of elementary axioms for the category of topological spaces and continuous mappings and to prove that any model of these axioms is equivalent to “the category of topological spaces” constructed over some model of Lawvere’s axioms. Furthermore, we prove that any complete category, model of the given axioms is equivalent to the category of topological spaces.