Much of the mathematics discussed thus far has probably seemed both familiar and useful. Topology is quite different. Most economists pass their lives quite innocent of its existence. And those who happen to make contact with this branch of mathematics will, likely as not, conclude that it is useless: a web of closely interwoven, almost circular definitions that don't seem to “do” anything. In fact topology is crucial to the development of a rigorous theory of competition. Without it we could not move past the definitions, notation, and examples of the preceding chapters to the theorems and proofs of the chapters which follow. For now you have to accept that on faith.

How then are we to come to grips with this subject? You need a better understanding of topology than a simple summary of the “main facts” would provide. On the other hand, attempting to compress a course in topology into the confines of a single chapter is clearly impossible, and probably worthless as well. What I have chosen to do instead is to “tell a story” about topology, to paint a mental picture of what it tries to accomplish and how. Don't try to absorb all of the details at once because that is scarcely possible and totally unnecessary. Try instead to grasp the main ideas, the broad outline of what topology is about. As we progress through the book and you see how topology is applied the subject will come to seem more and more natural.