TABLE OF CONTENTS Operators act on elements of sets. In appendix B, the set complement operator was defined with respect to a superset S. Furthermore, the definitions of open and closed sets were predicated on one definition: the open neighborhood. Now we are going to reverse things. An open neighborhood will be defined in terms of open sets, and a topological space will be defined in terms of its set of elements and its open sets. This appendix is meant to be introductory. See, e.g., [9] for a complete discussion of these topics.
A topological space is a set S together with a collection O of subsets called open sets such that
?, ? O and S ? O,
if U 1, U 2 ? O, then U 1 U 2 ? O,
the union of any collection of open sets is an open set.
Open sets can be arbitrarily designed as long as they satisfy the above three properties. The standard topology on 
has 
with O containing 
, the empty set ?, all open rectangles, and their unions. An example is the real line with open intervals, i.e., 
, with O consisting of any open interval, the union of open intervals, 
, and ?,.To show this we look to the three conditions in definition C.1.1:
, ?, ? O by definition,
any finite or infinite union of open...
