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Mathematical Supplement Appendix A Mathematical Supplement A.1 Elements of the Theory of Binary Relations A.1.1 Basic Operations on Sets Let us recall the definitions of some operations that are often performed on sets, and define the sets resulting from these operations. The union (or sum) of two sets A and B is the set [ A B consisting of all elements that belong to at least one of the sets A and B. The union of any number of sets is defined analogously: if we are given an arbitrary indexed family of sets, that is, a collection of sets Aa in which the index a runs through an arbitrary set (the index set), the union of the sets Aa is denoted by [ Aa : a By definition, this union consists of all elements that belong to at least one of the sets Aa . In addition, we agree that if the domain of the index a is not indicated, then the union extends over all the values that a may take in the given problem. The same refers to the operation of intersection. The intersection of two sets A and B is the set \ A B that consists of all elements belonging to both A and B. Analogously, the inter- section of an arbitrary (finite or infinite) number of sets Aa is the set \ Aa a 209 210 A Mathematical Supplement of all elements that belong to all of the sets Aa . The difference of the sets A and B is the set A n B consisting of those elements in A that are not contained in B. It is not assumed here that B ½ A. If B ½ A, then the set A n B is called the complement of the set B. It is clear that the complement to the set A n B is the set B itself. The following relations can be easily verified: (1) \ A n B = A n (A B): (2) If B ½ A (and only in this case) [ (A n B) B = A: (3) For any family of sets Aa the equation à ! [ \ [³ \ ´ Aa B = Aa B (A.1) a a (the distributive law) holds. In other words, the operation of intersection distributes over union. (4) à ! \ \ ³ [ ´ A = A B : (A.2) a a a a This is another form of the distributive law. (5) \ ³ \ ´ ³ \ ´ (A n B) C = A C n B C : S (6) If A = a Aa and B is an arbitrary set, then [ A n B = (A n B): a a (7) If the sets Ba are subsets of the set A, and Ca is their complement, that is, Ca = A n Ba , then [ \ \ [ A n Ba = Ca and A n Ba = Ca : (A.3) a a a a Two sets A and B are called disjoint (or nonintersecting) if their intersection is T empty: A B = ?. If some system of sets Aa is given and any two index sets belonging to this system and having different indices are disjoint, that is, A.1 Elements of the Theory of Binary Relations 211 \ Aa1 Aa2 = ? for a1 6= a2; then the sets Aa are said to be pairwise disjoint (or simply disjoint). It can be easily verified that the following auxiliary relations are correct: (i) If the sets Ai (i = 1;2;:::) form a decreasing sequence, that is, A1 ⊃ A2 ⊃ ¢¢¢ ⊃ Ai ⊃ ¢¢¢ ; and if \¥ Ai = ?; i=1 then [ [ [¥ A1 = (A1 n A2) (A2 n A3) ¢¢¢ = (Ai n Ai+1): i=1 It is obvious that the sets Ai n Ai+1 are disjoint. (ii) If Ai are disjoint sets and [¥ Bn = Ai; i=n+1 then \¥ Bn = ?: n=1 Moreover, it is obvious that the sets Bn form a decreasing sequence. (iii) If [¥ A = Ai; i=1 then [ [ [ [ [ i[¡1 [ A = A1 [A2 n A1] [A3 n (A1 A2)] ¢¢¢ [Ai n A j] ¢¢¢ j=1 [¥ i[¡1 = [Ai n A j] (A.4) i=1 j=1 Si¡1 Si¡1 (if i = 1 it is natural to put j=1 A j = ?). It is clear that the sets Ai n j=1 A j are disjoint. The formula (A.4) becomes simpler if the sets Ai form an increasing sequence, that is, 212 A Mathematical Supplement A1 ½ A2 ½ ¢¢¢ ½ Ai ½ ¢¢¢ : Then i[¡1 = Ai¡1 j=1 and formula (A.4) turns into [ [ [ [ A = A1 (A2 n A1) ¢¢¢ (Ai n Ai¡1) ¢¢¢ : (A.5) A.1.2 Topological Spaces Let E be a nonempty set the elements of which are called points. Definition A.1. The system S¤(x) of the subsets S(x) of the set E is called a system of neighborhoods, and its elements S(x) are called the neighborhoods of the point x if the following conditions are satisfied: Condition 1 Any neighborhood of the point x contains this point, that, is x 2 S(x). Condition 2 The intersection of any two neighborhoods of the point x is a neigh- borhood of the point x. Condition 3 Any set that contains a neighborhood of the point x is a neighbor- hood of the point x. Condition 4 Any neighborhood of the point x contains a set that is a neighbor- hood of each of its points as well as of the point x. Definition A.2. A point x is called an inner point of X if X is a neighborhood of x. If every neighborhood of x contains at least one point from X, then x is called a point of tangency of X. The set of all inner points of X is called the interior and is denoted by Int(X). The set of all tangency points of X is called the closure and is denoted by Cl(X). The set Fr(X) = Cl(X) n Int(X) is called the boundary (or frontier) of X. Definition A.3. The set X = Cl(X) is said to be closed. A set that is a neigh- borhood of every point (X = Int(X)) is said to be open. The system of all open subsets of the set E is called a topology of E and is denoted by T(E). The set E along with the topology (an ordered pair (E;T(E))) is called a topological space. Theorem A.1. The union of any number of open sets is an open set. The inter- section of a finite number of open sets is an open set. The space E and the empty set ? are open. A.1 Elements of the Theory of Binary Relations 213 Theorem A.2. The set X is closed if and only if its complement is open. The empty set ? and the set E are open and closed simultaneously. We have the following relations: [ [ Cl(X Y) = Cl(X) Cl(Y); ? = Int(?) = Cl(?); \ \ Int(X Y) = Int(X) Int(Y); E = Int(E) = Cl(E); Int(Int(X)) = Int(X) ½ X ½ Cl(X) = Cl(Cl(X)); \ Fr(X) = Cl(X) n Int(X) = Cl(X) Cl(C(X)) [ = Cl(Int(X) Int(Cl(X))): On a given set E one can define different systems of open sets, thus defining different topological spaces. A topology T1(E) is said to be stronger than T2(E) ¤ if T1(E) ⊃ T2(E). The discrete topology T (E) = expE is the strongest, while the topology T (E) = (E;?) is the weakest. 0 T Let E1 2 E, and let T(E1) = (E1 Xl : l 2 L) be the intersection of E1 with all open sets from E. The ordered pair (E1;T(E1)) can be considered as a topo- ¤ logical space. If ST(E)(x) is a system of neighborhoods of the point x in the topological space (E;T(E)), then the system of neighborhoods S¤ (x) of the T(E1) same point in the space (E1;T(E1)) is determined as the intersection of the el- ¤ ement from ST(E)(x) with the set E1. In this case the topology T(E1) is called ¤ the trace of the topology T(E) on E1, and S (x) is called the trace of the T(E1) neighborhoods of x on E1. Definition A.4. A topological space (E1;T(E1)), where T(E1) is the trace of the topology of the space (E;T(E)) on E1, is called a subspace of E. If it is necessary to indicate in which topology the sets are considered; the notation ClT (X), IntT (X);¶ FrT (X), etc. is used. Theorem A.3. The topology T1 is stronger than the topology T2 if and only if every neighborhood of an arbitrary point x 2 E in the topology T2 is a neighbor- hood of that point in the topology T1. Definition A.5. A system G¤ of open sets of the topological space (E;T) is called a basis of this space if every open set from T can be represented as a union of the sets from G¤. Definition A.6. The set X is called dense in the set Y if Cl(X) ¶ Y. If the set X is dense in E, then it is called everywhere dense. A closed set is called nowhere dense in E if its complement is everywhere dense. A topological space is called separable if it has a countable everywhere dense subset. 214 A Mathematical Supplement Definition A.7. If every pair of distinct points of the topological space (E;T) have neighborhoods that do not intersect, then the space is called a Hausdorff space. A Hausdorff space in which any two disjoint closed sets have disjoint neighborhoods is called normal. Definition A.8.
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