R. L. Moore |
Axioms for Topology |
[Just for amusement, here are the first two axioms from R. L. Moore's (in)famous book on point set topology. Axiom 0 is stated on page 1 of his book, without motivation, although a 3 page introduction clarifies some of the notation. Axiom 1 appears on pages 1 and 2 with no motivation at all. The quotes from Moore's book are in boldface below. R. L. Moore, Foundations of Point Set Theory, Revised Edition, American Mathematical Society Colloquium Publications, Vol. XIII, 1962. (First edition published 1932.)
I thought about Axiom 1 for a while and still found it completely incomprehesible.]Undefined notions are point and region.
AXIOM 0. Every region is a point set.
[Various definitions, including that of limit point.]
NOTATION. Hereafter, throughout this book, the letter S will be used to denote the set to which X belongs if and only if X is a point. If M is a point set, the notation M will be used to designate a point set such that X belongs to it if and only if X either belongs to M or X is a limit point of M. The point set M will sometimes be called the closure of M.
AXIOM 1. There exists a sequence G1 , G2 , G3 , . . . such that (1) for each n, Gn is a collection covering S such that each element of Gn is a region, (2) for each n, Gn+1 is a subcollection of Gn , (3) if R is a region, X is a point of R and Y is a point of R, either identical with X or not, then there exists a natural number m such that if g is any region belonging to the collection Gm and containing X then g is a subset of R and, unless Y is X, g does not contain Y, (4) if M1 , M2 , M3 , . . . is a sequence of closed point sets such that, for each n, Mn contains Mn+1 and, for each n, there exists a region gn of the collection Gn such that Mn is a subset of gn, then there is at least one point common to all the point sets of the sequence M1 , M2 , M3 , . . ..
