Z orn ’s lemma. Suppose S, < is a partially ordered set with the property that every chain in S has an upper bound. Then S contains amaximal element.

The axiom o f choice. Suppose {Si}, i E I, is a family of nonemptysets. Then there is a function / from I into U / Si such that f(i) E Sifor each i e I.

opology has reached the point where a mathematician engagedin topological research is not only justified in calling himself a topologist,but he must specify whether he is a point set topologist, differentialtopologist, algebraic topologist, or some other topological specialist.

Geometrically, topologywas the study of properties preserved by a certain group of transformations, the homeomorphisms. Geometry itself can be considered as thestudy of properties preserved by certain types of functions; e.g., Euclideanmetric geometry is the study of properties preserved by rigid (that is,distance-preserving) transformations (known sometimes as congruences).(Of course, as with topology, it is somewhat unfair to try to define geometry as the study of one particular thing.)