Students must prove all of the theorems in this undergraduate-level text, which features extensive outlines to assist in study and comprehension. Thorough and well-written, the treatment provides sufficient material for a one-year undergraduate course. The logical presentation anticipates students' questions, and complete definitions and expositions of topics relate new concepts to previously discussed subjects. Most of the material focuses on point-set topology with the exception of the last chapter. Topics include sets and functions, infinite sets and transfinite numbers, topological spaces and basic concepts, product spaces, connectivity, and compactness. Additional subjects include separation axioms, complete spaces, and homotopy and the fundamental group. Numerous hints and figures illuminate the text.
You have certainly had much experience by now with both of the concepts in the title of this chapter. But it is a fact that a thorough familiarity with sets and functions is essential to any understanding of modern mathematics, so the material in this chapter is of basic importance to all of your future work. You should read it carefully, do all of the exercises, and prove all of the theorems.
1. SETS
You may recall that, in Euclidean geometry, the words âpointâ and âlineâ are not defined. Instead, it is assumed that everyone has a good idea of what points and lines are, and we let it go at that. This failure to define terms is not because of laziness or sloppiness, though; the fact is that in any system of human thought we have to start somewhere, so there must always be some primitive notions that cannot be defined because we have nothing to define them in terms of.
In this course, the most basic notions that we will use are those of set, element of a set, and what it means for a given element to belong to a given set: we will not define any of these ideas. But of course it is all clear: a set is a collection of objects, an element of the set is one of these objects, and it is always clear when a given object is an element of a given set. For example, the set of all people who are now citizens of the United States is the collection of all those people, and only those people, who are now citizens of the United States. The set is the collection of all U. S. citizens, an element of the set is a U. S. citizen, and one can always determine whether a given object belongs to the set: the given object must be a person and that person must be a U. S. citizen.
Two sets are the same if and only if they have exactly the same elements. Formally,
1.1. Definition. Let A and B be sets. Then
1) A is a subset of B, written A â B, if every element of A is also an element of B.
2) A and B are equal, written A = B, if both A â B and B â A.
Notice that A â B does not imply that A and B cannot be equal. If A â B and A is not equal to B (in which case we call A a proper subset of B), we sometimes write A
B, for emphasis.
A word about definitions is in order. A definition in mathematics is always ...
