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Appendix a Prerequisites to Set Theory and General Topology

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Appendix a Prerequisites to Set Theory and General Topology Appendix A Prerequisites to Set Theory and General Topology 1. General remarks As mentioned in the Introduction, this book is intended to the reader acquainted with the basics of "naive" set theory. The modern (axiomatic and "model-theoretic") set theory is not evoked in fact.l We advice the reader to peruse the Introduction to the book "Topology" by K. Kuratowski. It is also helpful to consult other sources; for instance, the celebrated books by F. Hausdorff, P. S. Aleksandroff, or 1. P. Natanson as regards ordinal numbers and transfinite induction. A few remarks about notation and terminology are now in order. We use the following symbols of set-theoretic operations and relations: u,n, \, E,~, C,:J . We abstain from using the notations like ~, on presuming that the symbol C does not exclude the equality of sets. The empty set is denoted by 0; the cardinality of E, by cardE. The class of all x satisfying some property is denoted by {xl·· .}. To denote a function2 we usually take a single letter: f,g,r.p and so on. In this event f(x} is the value of f at a point x; the parentheses are sometimes omitted. We practice only the definition of mappings by the formulas of the type x --+ f(x}. At last, to denote a function on some set A, we use the record like {X"'}"'EA. It this case we speak of a family € == {Xa}aEA and call A the "index set." A particular case of a family is a simple sequence {Xn}~=l (the role of A is played by the set of natural numbers). The meaning of the terms "system," "totality," "collection," "tuple," etc. is always clear from the context (we usually imply the set of members of some family). lPor instance, cf. T. Jech [1) and E. I. Gordon and S. P. Morozov [1). 2In this book the words "function," ''mapping,'' and "operator" are viewed as synonyms. 563 564 BOOLEAN ALGEBRAS IN ANALYSIS The inverse of a function f is denoted by f~l. Irrespective of whether or not the inverse of a function f exists, the symbol f~l(e) always stands for the inverse image of a set e as well as f(e) symbolizes the image of e. The restriction of a function f to some e is denoted by fie. Assume given a family of sets {Eah'EA. The product of this family is the set of all families c == {ea}aEA satisfying eo E Ea for every 0: E A. This product is denoted by ITaEA Ea. In the case when A = {I, 2}, we obtain the product of two sets which is denoted by El X E2; this is the collection of all ordered pairs (el' e2), where el EEl and e2 E E2. If El = E2 = E then we write E2 instead of El x E2 and speak about the "product of two copies of E"; the analogous terminology applies to the general case in which all Be, (0: E A) coincide. To each value of the index 0:0 E A we assign the "projection" ?rao that sends an arbitrary family c == {ea} treated as a point in the product Ea to the element The values of these mappings are called the "projections" or "coordinates" of c. 2. Partially ordered sets A partial ordering or partial order3 on a nonempty set f£ is a subset P C f£2 satisfying the following axioms: I. (x,x) E P for all x E f£. II. If (x,y) E P and (y,x) E P then x = y. III. If (x, y) E P and (y, z) E P then (x, z) E P. Thus, a partial order is a RELATION between the elements of f£. The axioms I and III express the REFLEXIVITY and TRANSITIVITY of this relation; the axiom II tells US that it is ANTISYMMETRIC. As a rule, we write x ~ y or y 2: x rather than (x, y) E P. Other similar symbols may replace ~; for instance, -<. The axioms I~III may be rewritten as I. x ~ x for all x E f£. II. If x ~ y and y ~ x then x = y. III. If x ~ y and y ~ z then x ~ z. The formula x < y (or y > x) means by definition that x ~ y and x -I- y. The expressions of the form a ~ b, a < b, etc. are called inequalities. A partially ordered set or poset is some set f£ furnished with some partial order P on f£; i.e., the pair {f£, P}. Most often we denote a partially ordered set by the same letter f£ as the underlying set; accordingly, we call the members of f£ "elements of a partially ordered set." This abuse of the language is very common in mathematics but is excusable only in the case when f£ is equipped with a single order. If every two elements in a partially ordered set f£ are compatible, Le., either (x,y) E P or (y,x) E P for all x,y E f£; then we say that f£ is linearly ordered. As an example, we may take every set of real numbers furnished with the conventional order. Likewise on the real axis, the set of all x satisfying the inequality a ~ x ~ b is called an interval and we denote it by [a, b]. 3The word "partial" is often omitted. APPENDIX A: PrereIJ.uisites to Set Theory andGeneral Topology 565 Let !£ be equipped with some partial order. Each subset !£o c !£ may also be furnished with some partial order; to this end, we put It is easy to see that the axioms I-III are satisfied. The so-defined partial order Po is said to be induced by P or induced from outside. Practically speaking, this means that all inequalities have the same sense in !£o as they have in !£. It might happen that !£o is linearly ordered in the induced order; in this event we call &:0 a chain. An element Xo of a partially ordered set !£ is maximal provided that the inequality x ~ Xo implies x = Xo. A significant role in mathematics belongs to the following Kuratowski-Zorn Lemma.4 Assume that a partially ordered set !£ possesses the following property: to each chain !£o c !£ there is some y E !£ such that x ~ y for all x E !£o. Then, to whatever element x E !£, there is a maximal element xo satisfying the inequality Xo ~ x. The Kuratowski-Zorn Lemma often replaces the principle of mathematical induc­ tion in proofs, enabling us to avoid ordinal numbers. However, it is sometimes more natural to use the Zermelo Theorem or transfinite numbers which leads faster to the aim (for instance, cf. the proof of the theorem about the structure of a homogeneous BA in Chapter 9 of this book). 3. Topologies An important example of a partially ordered set is provided by the system of topologies 5 on some set R. By a topology we routinely mean a class T of subsets of R that is closed under all unions and finite intersections. The members of T are called open and their comple­ ments, closed sets in R. The system T' of all closed sets also uniquely determined the topology of Rj the role of the latter may be performed by each class of sets closed un­ der all intersections and finite unions. The pair {R, T} is a topological space denoted by the same letter R. We say that a topology Tl is stronger than a topology T2 whenever Tl J T2 (or, which is the same, T{ J Tn. In this event we say also that Tl majorizes, or dominates 7"2, or T2 is weaker than Tl, etc. For each nonempty set T of topologies on R, there always exists a unique WEAKEST topology among those dominating every member of T; in much the same way, there is a unique STRONGEST topology among those dominated by every member of T. We say that x is an interior point of a set V or that V is a neighborhood about x whenever there is an open set G satisfying x E G C V. Let W(x) stand for the collection of all neighborhoods of x. If Wo is such a set of neighborhoods that to each V E W(x) there is some Vo E Wo satisfying Vo C V then we say that Wo is a base of neighborhoods about x. Assume that to each point x E R there corresponds some base of neighborhoods Wo(x)j then the inclusion GET means that to each point x E G there is some V E Wo{x) satisfying V c G. 4K. Kuratowski [1] and M. Zorn [1]. This proposition is equivalent to the axiom of choice and also to the so-called "Hausdorff maximality principle" (J. Kelley [2]). 5N. Bourbaki [1]; K. Kuratowski [2]; and J. Kelley [2]. 566 BOOLEAN ALGEBRAS IN ANALYSIS We may thus uniquely recover the topology 7 from available bases of neighbor­ hoods about all points. This idea is often used for the INITIAL INTRODUCTION OF A TOPOLOGY. Assume given a family {!Uo(x)} whose every member !Uo(x) is some class of subsets containing the point x. Assume further that the following hold: I. If V!, V2 E !Uo(x) then there is a set V E !Uo(x) lying in Vi n V2. II. Given V E !Uo(x), we may find V' E !Uo(x) so that each class !Uo(x/), x' E V', contains at least one set included in V.
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