Just two years after David Hilbert published his Grundlagen der Geometrie (1899), Eliakim Hastings Moore was teaching a course from the book to his graduate students at the University of Chicago. Among his auditors, Oswald Veblen quickly embraced the axiomatic methods inherent in Hilberta**s work, while both he and Moore encouraged yet another Chicago student, Robert L. Moore, to engage in axiomatic research. At Princeton, Veblen employed this a**from the ground upa** approach first in projective geometry and then in combinatorial topology, while R. L. Moore first at the University of Pennsylvania and then at the University of Texas, Austin, encouraged a generation of American point-set topologists to approach their chosen subfield from this particular point of view.

This talk will explore how this emphasis on an axiomatic mode of thought allowed members of the American mathematical research community to carve out research agendas in each of these two brands of topology and, in so doing, to begin to establish the United States as a player in the international mathematical arena.