| topology |
| 1 | Logic, Set Theory, and the Axiom of Choice Our purpose in this introductory chapter is to review some of the foundation concepts, on which the remainder of the text rests. We assume that someone who wishes to learn topology has some familiarity with all of these ideas, but we also include here some fairly extensive references so that a prospective student may fill in, on his own, any gaps in his knowledge. |
| 2 | A basic tool in mathematics, at virtually any level, is logic. While logic has been developed into an elaborate subject of its own, often under the name of symbolic logic or mathematical logic, the logic which we need here deals mostly with the basic rules for manipulation of mathematical statements. This is a sensible place to begin our review; we indicate some of the more fancy treatments in the References at the end of the chapter. |
| 3 | One traditionally refers to mathematical statements by capital letters, such as P, Q, etc. Thus P can refer to things like "49 is a positive number," "there are infinitely many points in the plane," "the area of a circle is wr 2 ," etc. As a matter of convention, when one writes a mathematical statement, by itself, we mean to say that the statement is true. For example, if we wrote "Professor Schmitt has proved P in his last lecture. |
| 4 | Hence, we con- clude Q," we really mean that Schmitt has shown that P is true and thus we conclude (somehow, presumably by other remarks) that statement Q is true. Or another way this may arise is that we give the name P to some statement; after several lines, we show that P is true, but we write, for short, something like "Hence P." On the other hand, when mathematical statements or propositions occur in conjunction with other statements, they may or may not be true. |
| 5 | The 1 2 Topology standard sort of expression, which occurs all the time, is "IfP, then Q." This really means, if P is true, then Q is true. It is an accepted convention that when P is false, the total assertion "If P, then Q" is regarded as true because if P is not true, we need not worry about the truth of the claim "if P is true, then Q is true." For example, the ridiculous assertion "If 2 + 2 = 5, then 3 + 3 = 7" is regarded as correct or true, because since 2 + 2 is not 5, it doesn't matter what 3 + 3 is. |
| 6 | A standard and convenient notation is ~P, meaning not P or the nega- tion of P. If P means x is greater than 3, then ~P would mean x is less than or equal to 3. Standing by itself, ~P would mean that P is false. Clearly, <~~P means P. Various notions have become accepted. For example, "P and Q" means that both P and Q are true. "P or Q" means that at least one of the two statements is true. It is important to note that when we say "P or Q" we do allow the possibility that both are true. |
| 7 | If we wish to exclude this case, we would write "P or Q" but not "P and Q." One can now express some of these concepts in simpler and more concise ways. As a simple illustration, "P or Q" is precisely the same thing as <~(~P and ~Q), because "P or Q" encompasses all possibilities except that both P and Q are false. As a deeper application of this idea, we can rewrite "if P, then Q" as ~(P and ~Q), because it really means that if P is true, so is Q, which is the same thing as saying that it is not the case that if P is true, Q is false. |
| 8 | The statement "if P, then Q" occurs in various costumes all of which mean the same thing: "when P is true, so is Q" "P implies Q" "under the hypothesis P, we have the conclusion Q" "P is a sufficient condition for Q" etc. We will use the standard shorthand "P =» Q" for "if P, then Q." Note that if "P => Q" is true, it need not follow that "Q => P" is true. If P =» Q and Q=* P, then we say that P and Q are equivalent (the being both true or both false). |
| 9 | This is usually written P = Q and is sometimes referred to as "a necessary and sufficient condition for Q is P," or "P if and only if Q." In more complicated situations, P, Q, etc., depend on variables and are not absolute statements. For example, "x > 5" is a statement which is true when x is bigger than 5 but is false otherwise. In general, the depend- Logic, Set Theory, Axiom of Choice 3 ency of statements on variables means that things may or may not be true, but it does not affect the rules of logic which deal with manipulating statements. |
| 10 | The basic rules, which may be easily checked, are the following: 1) P=s>P 2) If P => Q and Q => R, then P => R. 3) P =» Q is equivalent to ~Q => ~P. 4) ~(P and ~P). If we wanted to prove 3) we would show that a) if P => Q, then ~Q =» ~P and b) if ~Q =» ~P, then P=*Q. For illustration, we write out the proof of a). We suppose P => Q. We wish to show that if ~Q is true, then ~P is true. So we assume also that ~Q is true or Q is false. |
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