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General Topology

Dover Publications, Inc.

PRELIMINARIES

The only prerequisites for understanding this book are a knowledge of a few of the properties of the real numbers and a reasonable endowment of that invaluable quality, mathematical maturity. All of the definitions and basic theorems which are assumed later are collected in this first chapter. The treatment is reasonably self-contained, but, especially in the discussion of the number system, a good many details are omitted. The most profound results of the chapter are theorems of set theory, of which a systematic treatment is given in the appendix. Because the chapter is intended primarily for reference it is suggested that the reader review the first two sections and then turn to chapter one, using the remainder of the chapter if need arises. Many of the definitions are repeated when they first occur in the course of the work.

SETS

We shall be concerned with sets and with members of sets. "Set," "class," "family," "collection," and "aggregate" are synonymous, and the symbol ε denotes membership. Thus x ε A if and only if x is a member (an element, a point) of A. Two sets are identical iff they have the same members, and equality is always used to mean identity. Consequently, A = B if and only if, for each xy x ε A when and only when x ε B.

Sets will be formed by means of braces, so that {x: ... (proposition about x) ...} is the set of all points x such that the proposition about x is correct. Schematically, y ε {x: ... (proposition about x) ...} if and only if the corresponding proposition about y is correct. For example, if A is a set, then y ε {x: x ε A} iff y ε A. Because sets having the same members are identical, A = {x: x ε A)y a pleasant if not astonishing fact. It is to be understood that in this scheme for constructing sets is a dummy variable, in the sense that we may replace it by any other variable that does not occur in the proposition. Thus {x: x ε A) = [y: y ε A)y but {x: x ε A) ≠ {A: A ε A}.

There is a very useful rule about the construction of sets in this fashion. If sets are constructed from two different propositions by the use of the convention above, and if the two propositions are logically equivalent, then the constructed sets are identical. The rule may be justified by showing that the constructed sets have the same members. For example, if A and B are sets, then {x: x ε A or x ε B) = {x: x ε B or x ε A)y because y belongs to the first iff y ε A or y ε B, and this is the case iff y ε B or y ε Ay which is correct iff y is a member of the second set. All of the theorems of the next section are proved in precisely this way.

SUBSETS AND COMPLEMENTS; UNION AND INTERSECTION

If A and B are sets (or families, or collections), then A is a subset (subfamily, subcollection) of B if and only if each member of A is a member of B. In this case we also say that A is contained in B and that B contains Ay and we write the following: A [subset] B and B [contains] Z) A. Thus A [subset] B iff for each x it is true that x ε B whenever x ε A. The set A is a proper subset of B (A is properly contained in B and B properly contains A) iff A [subset] B and A ≠ B. If A is a subset of B and B is a subset of C, then clearly A is a subset of C. If A [subset] B and B [subset] A, then A = By for in this case each member of A is a member of B and conversely.

The union (sum, logical sum, join) of the sets A and By written A [union] By is the set of all points which belong either to A or to B; that is, A [union] B = {x: x ε A or x ε B}. It is understood that "or" is used here (and always) in the non-exclusive sense, and that points which belong to both A and B also belong to A [union] B. The intersection (product, meet) of sets A and By written A [intersection] B, is the set of all points which belong to both A and B; that is, A [intersection] B = {x: x ε A and x ε B}. The void set (empty set) is denoted 0 and is defined to be {x: x ≠ x}. (Any proposition which is always false could be used here instead of x ≠ x.) The void set is a subset of every set A because each member of 0 (there are none) belongs to A. The inclusions, 0 [subset] A [intersection] B [subset] A [subset] A [union] B, are valid for every pair of sets A and B. Two sets A and B are disjoint, or non-intersecting, iff A [intersection] B = 0; that is, no member of A is also a member of B. The sets A and B intersect iff there is a point which belongs to both, so that A [intersection] B ≠ 0. If a is a family of sets (the members of (a are sets), then a is a disjoint family iff no two members of a intersect.

The absolute complement of a set Ay written ~ Ay is {x: x [??] A}. The relative complement of A with respect to a set X is X [intersection] ~ Ay or simply X ~ A. This set is also called the difference of X and A. For each set A it is true that ~~ A = A; the corresponding statement for relative complements is slightly more complicated and is given as part of 0.2.

One must distinguish very carefully between "member" and "subset." The set whose only member is x is called singleton x and is denoted {x}. Observe that {0} is not void, since 0 ε {0}, and hence 0 ≠ {0}. In general, x ε A if and only if {x} a A.

The two following theorems, of which we prove only a part, state some of the most commonly used relationships between the various definitions given above. These are basic facts and will frequently be used without explicit reference.

1 Theorem Let A and B be subsets of a set X. Then A [subset] B if and only if any one of the following conditions holds:

A [intersection] B = A, B = A [union] B, X ~ B [subset] X ~ A, A [intersection] X ~ B = 0, or (X ~ a) B = X.

2 Theorem Let A, B, C, and X be sets. Then:

(a) X ~ (X ~ A) = A [intersection] X.

(b) (Commutative laws) A [union] B = B [union] A and A [intersection] B = B [intersection] A.

(c) (Associative laws) A [union] (B [union] C) = (A [union] B) [union] C and A [intersection] (B [intersection] C) = (A [intersection] B) [intersection] C.

(d) (Distributive laws) A [intersection] (B [union] C) = (A [intersection] B) [union] (A [intersection] C) and A [union] (B [intersection] C) = (A [union] B) [intersection] (A [union] C).

(e) (De Morgan formulae) X ~ (A [union] B) = (X ~ A) [intersection] (X ~ B) and X ~ (A [intersection] B) = (X ~ A) [union] (X ~ B).

PROOF Proof of (a): A point x is a member of X ~ (X ~ A) iff x ε X and x [not epsilon] X ~ A. Since x [not epsilon] X ~ A iff x [not epsilon] X or x ε A, it follows that x ε X ~ (X ~ A) iff x ε X and either x [not epsilon] or x ε A. The first of these alternatives is impossible, so that x ε X ~ (X ~ A) iff x ε X and x ε A; that is, iff x ε X [intersection] A. Hence X ~ (X ~ A) = A [intersection] X. Proof of first part of (d): A point x is a member of A [intersection] (B [union] C) iff x εA and either x ε B or x ε C. This is the case iff either x belongs to both A and B or x belongs to both A and C. Hence x ε A [intersection] (B [union] C) iff x ε (A [intersection] B) [union] (A [intersection] C), and equality is proved. |

If A1, A2, ..., An are sets, then A1 [union] A2 [union] ... [union] An is the union of the sets and A1 [intersection] A2 [intersection] ... [intersection] An is their intersection. It does not matter how the terms are grouped in computing the union or intersection because of the associative laws. We shall also have to consider the union of the members of non-finite families of sets and it is extremely convenient to have a notation for this union. Consider the following situation: for each member a of a set A, which we call an index set, we suppose that a set Xa is given. Then the union of all the Xa, denoted [union] [Xa: a ε A}, is defined to be the set of all points x such that x ε Xa for some a in A. In a similar way the intersection of all Xa for a in A, denoted [union] {Xa: a ε A}, is defined to be {x: x ε Xa for each a in A). A very important special case arises when the index set is itself a family a of sets and XA is the set A for each A in a. Then the foregoing definitions become: [union] {A: A ε a} = {x: x [psilon] A for some A in a} and [intersection] {A: A ε a} = x: x ε A for each A in a}.

There are many theorems of an algebraic character on the union and intersection of the members of families of sets, but we shall need only the following, the proof of which is omitted.

3 Theorem Let A be an index set, and for each a in A let Xa be a subset of a fixed set Y. Then:

(a) If B is a subset of A, then [union] {Xb: b ε B] [subset] {Xa a ε A} and [intersection] {Xb: b ε B} [contains] [intersection] {Xa: a ε A}.

(b) (De Morgan formulae) Y ~ [union] {Xa: a ε A} = [Intersection] {Y ~ Xa: a ε A] and Y ~ [intersection] {Xa: a ε A} = [union] {Y ~ Xa: a ε A}.

The De Morgan formulae are usually stated in the abbreviated form: the complement of the union is the intersection of the complements, and the complement of an intersection is the union of the complements.

It should be emphasized that a reasonable facility with this sort of set theoretic computation is essential. The appendix contains a long list of theorems which are recommended as exercises for the beginning student. (See the section on elementary algebra of classes.)

4Notes In most of the early work on set theory the union of two sets A and B was denoted by A + B and the intersection by AB, in analogy with the usual operations on the real numbers. Some of the same algebraic laws do hold; however, there is compelling reason for not following this usage. Frequently set theoretic calculations are made in a group, a field, or a linear space. If A and B are subsets of an (additively written) group, then {c: c = a + b for some a in A and some b in B} is a natural candidate for the label "A + B" and it is natural to denote {x: - x ε A} by -A. Since the sets just described are used systematically in calculations where union, intersection, and complement also appear, the choice of notation made here seems the most reasonable.

The notation used here for construction of sets is the one most widely used today, but "E" for 'The set of all * such that" is X also used. The critical feature of a notation of this sort is the following: one must be sure just which is the dummy variable. An example will clarify this contention. The set of all squares of positive numbers might be denoted quite naturally by x2 > X > 0}, and, proceeding, [{x2 + a2: x< 1 + 2a} also has a natural meaning. Unfortunately, the latter has three possible natural meanings, namely: {z: for some x and some ay z = x2 + a2 and x< 1 + 2a]y {z: for some xy z = x2 + a2 and x< 1 + 2a]y and {z: for some ay z = x2 + a2 and x < 1 + 2a}. These sets are quite different, for the first depends on neither x nor a, the second is dependent on ay and the third depends on x. In slightly more technical terms one says that "x" and "a" are both dummies in the first, "x" is a dummy in the second, and "a" in the third. To avoid ambiguity, in each use of the brace notation the first space after the brace and preceding the colon is always occupied by the dummy variable.

Finally, it is interesting to consider one other notational feature. In reading such expressions as "A [intersection] {B [union] C)" the parentheses are essential. However, this could have been avoided by a slightly different choice of notation. If we had used "U AB" instead of "A [union] B," and similarly for intersection, then all parentheses could be omitted. (This general method of avoiding parentheses is well known in mathematical logic.) In the modified notation the first distributive law and the associative law for unions would then be stated: [intersection] A [union] BC = [union] [intersection] AB [intersection] AC and A [union] BC — [union] [union]ABC. The shorthand notation also reads well; for example, [union] AB is the union of A and B.

RELATIONS

The notion of set has been taken as basic in this treatment, and we are therefore faced with the task of defining other necessary concepts in terms of sets. In particular, the notions of ordering and function must be defined. It turns out that these may be treated as relations, and that relations can be defined rather naturally as sets having a certain special structure. This section is therefore devoted to a brief statement of the definitions and elementary theorems of the algebra of relations.

Suppose that we are given a relation (in the intuitive sense) between certain pairs of objects. The basic idea is that the relation may be represented as the set of all pairs of mutually related objects. For example, the set of all pairs consisting of a number and its cube might be called the cubing relation. Of course, in order to use this method of realization it is necessary that we have available the notion of ordered pair. This notion can be defined in terms of sets. The basic facts which we need here are: each ordered pair has a first coordinate and a second coordinate, and two ordered pairs are equal (identical) if and only if they have the same first coordinate and the same second coordinate. The ordered pair with first coordinate x and second coordinate y is denoted (x, y). Thus (x,y) = (u,v) if and only if x = u and y = v.

It is convenient to extend the device for the formation of sets so that {(x,y): ...} is the set of all pairs (x,y) such that .... This convention is not strictly necessary, for the same set is obtained by the specification: {z: for some x and some y, z = (x,y) and ...}.

A relation is a set of ordered pairs; that is, a relation is a set, each member of which is an ordered pair. If R is a relation we write xRy and {x,y) ε R interchangeably, and we say that x is R-related toy if and only if xRy. The domain of a relation R is the set of all first coordinates of members of Ry and its range is the set of all second coordinates. Formally, domain R = {x: for some y, (x,y) ε R} and range R — {y: for some x, (x,y) ε R}. One of the simplest relations is the set of all pairs (x,y) such that x is a member of some fixed set A and y is a member of some fixed set B. This relation is the cartesian product of A and B and is denoted by A X B. Thus A X B = {(x,y): x ε A and y ε B}. If B is non-void the domain of A X B is A. It is evident that every relation is a subset of the cartesian product of its domain and range.

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Excerpted from General Topology by John L. Kelley. Copyright © 2017 John L. Kelley. Excerpted by permission of Dover Publications, Inc..
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