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Appendix a Set Theory Appendix A Set Theory A.1 Sets The purpose of this appendix is to introduce some of the basic ideas and terminologies from set theory which are essential to our present work. In this naive treatment, we do not intend to give a complete and precise analysis of set theory, which belongs to the foundations of mathematics and to mathematical logic. Rather, we shall deal with sets on an intuitive basis. We remark that this usage can be formally justified. Intuitively, a set is a collection of objects called members of the set. The terms “collection” and “family” are used as synonyms for “set”. If an object x is a member of a set X, we write x ∈ X to express this fact. Implicit in the idea of a set is the notion that a given object either belongs or does not belong to the set. The statement “x is not a member of X” is indicated by x ∈/ X. The objects which make up a set are usually called the elements or points of the set. Given two sets A and X, we say that A is a subset of X if every element of A is also an element of X. In this case, we write A ⊆ X (read A is contained in X)or X ⊇ A (read X contains A). The sets A and X are said to be equal, written A = X, if A ⊆ X and X ⊆ A. We call A a proper subset of X (written A ⊂ X)ifA ⊆ X but A = X.Theempty or null set which has no element is denoted by ∅.Wehave the inclusion ∅ ⊆ X for every set X. The set whose only element is x is called a singleton, denoted by {x}. Note that ∅ ={∅}.IfP (x) is a statement about elements in X, that is either true or false for a given element of X, then the subset of all the x ∈ X for which P (x) is true is denoted by {x ∈ X | P (x)} or {x ∈ X : P (x)}. In a particular mathematical discussion, there is usually a set which consists of all primary elements under consideration. This set is referred to as the “universe”. To avoid any logical difficulties, all the sets we consider in this section are assumed to be subsets of the universe X. The difference of two sets A and B, denoted by A − B,istheset{x ∈ A | x ∈/ B}.If B ⊆ A,thecomplement of B in A is A − B. Notice that the complement operation is defined only when one set is contained in the other, whereas the difference operation does not have such a restriction. The union of two sets A and B is the set A ∪ B = © Springer Nature Singapore Pte Ltd. 2019 411 T. B. Singh, Introduction to Topology, https://doi.org/10.1007/978-981-13-6954-4 412 Appendix A: Set Theory {x | x belongs to at least one ofA, B}.Theintersection of two sets A and B is the set A ∩ B ={x | xbelongs to both A and B}. When A ∩ B = ∅,thesetsA and B are called disjoint; otherwise we say that they intersect. Proposition A.1.1 (a) X − (X − A) = A. (b) B ⊆ A ⇔ X − A ⊆ X − B. (c) A = B ⇔ X − A = X − B. Proposition A.1.2 A ∪ A = A ∩ A. Proposition A.1.3 A ∪ B = B ∪ A, A ∩ B = B ∩ A. Proposition A.1.4 A ⊆ B ⇔ A = A ∩ B ⇔ B = A ∪ B. Proposition A.1.5 (a) A ∪ (B ∪ C) = (A ∪ B) ∪ C. (b) A ∩ (B ∩ C) = (A ∩ B) ∩ C. Proposition A.1.6 (a) A ∩ (B ∪ C) = (A ∩ B) ∪ (∩C). (b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Proposition A.1.7 (a) A − B = A ∩ (X − B). X − (A ∪ B) = (X − A) ∩ (X − B) (b) (De Morgan’s laws). X − (A ∩ B) = (X − A) ∪ (X − B) (c) (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B). (d) If X = A ∪ B and A ∩ B = ∅, then B = X − A. Let J be a nonempty set, and suppose that a set A j is given for each j ∈ J. Then the collection of sets {A j | j ∈ J}, also written as {A j } j∈J , is called an indexed family of sets, and J is called an indexing set for the family. The union of an indexed family {A j | j ∈ J} of subsets of a set X is the set A j ={x ∈ X | x ∈ A j for some j in J}, j∈J and the intersection is the set A j ={x ∈ X | x ∈ A j for every j in J}. j∈J { | ∈ } The union of the sets A j is also denoted by A j j J , and their intersection by { | ∈ } A j j J . If there is no ambiguity about the indexing set, we simply use A j ={ ,..., } > for the union and A j for the intersection. If J 1 n , n 0 an integer, then n ,..., n we write j=1 A j for the union of A1 An, and j=1 A j for their intersection. Observe that any nonempty collection C of sets can be considered an indexed family of sets by “self-indexing”: The indexing set is C itself and one assigns to each S ∈ C the set S. Accordingly, the foregoing definitions become Appendix A: Set Theory 413 {S : S ∈ C } = {x | x ∈ S for some S in C } and {S : S ∈ C } = {x | x ∈ S for every S in C }. If we allow the collection C to be the empty set, then, by convention, {S : S ∈ C } = ∅ and {S : S ∈ C } = X, the specified universe of the discourse. Proposition A.1.8 Let A j ,j∈ J, be a family of subsets of a set X. Then we have − = − − = − (a) X j A j j X A j , and X j A j j X A j . ⊂ ⊆ ⊇ (b) If K J, then k∈K Ak j∈J A j and k∈K Ak j∈J A j . A.2 Functions With each two objects x, y, there corresponds a new object (x, y), called their ordered pair. This is another primitive notion that we will use without a formal definition. Ordered pairs are subject to the condition: (x, y) = x , y ⇔ x = x and y = y . Accordingly, (x, y) = (y, x) ⇔ x = y. The first (resp. second) element of an ordered pair is called the first (resp. second) coordinate. Given two sets X1 and X2, their Cartesian product X1 × X2 is defined to be the set of all ordered pairs (x1, x2), where x j ∈ X j for j = 1, 2. Thus X1 × X2 ={(x1, x2) | x j ∈ X j for every j = 1, 2}.Note that X1 × X2 = ∅ if and only if X1 = ∅ or X2 = ∅. When both X1 and X2 are nonempty, X1 × X2 = X2 × X1 ⇔ X1 = X2. Let X and Y be two sets. A function f from the set X to Y (written f : X → Y ) is a subset of X × Y with the following property: for each x ∈ X, there is one and only one y ∈ Y such that (x, y) ∈ f. A function is also referred to as a mapping (or briefly, a map). We write y = f (x) to denote (x, y) ∈ f, and say that y is the image of x under f or the value of f at x. We also say that f maps (or carries) x into y or f sends (or takes) x to y. A function f from X to Y is usually defined by specifying its value at each x ∈ X, and if the value at a typical point x ∈ X is f (x), we write x → f (x) to give f . We refer to X as the domain, and Y as the codomain of f.The set f (X) ={f (x) | x ∈ X}, also denoted by im( f ), is referred to as the range of f. The identity function on X, which sends every element of X to itself, is denoted by 1 or 1X .Amapc : X → Y which sends every element of X to a single element of Y is called a constant function. Notice that the range of a constant function consists of just one element. If A ⊂ X, the function i : A → X, a → a, is called the inclusion map of A into X.If f : X → Y and A ⊂ X, then the restriction of f to A is the function f |A : A → Y defined by ( f |A) (a) = f (a) ∀ a ∈ A. The other way around, if A ⊂ X and g : A → Y is a function, then an extension of g over X is a function G : X → Y such that G|A = g. The inclusion map i : A → X is the restriction of the identity map 1 on X to A. { | ∈ } Proposition A.2.1 Let X be a set and A j j J a family of subsets of X with X = A j . If, for each j ∈ J, fj : A j → Y is a function such that f j | A j ∩ Ak = 414 Appendix A: Set Theory fk | A j ∩ Ak for all j, k ∈ J, then there exists a unique function F : X → Ywhich extends each f j . Proof Given x ∈ X, there exists an index j ∈ J such that x ∈ A j . We put F (x) = f j (x) if x ∈ A j .Ifx ∈ A j ∩ Ak , then f j (x) = fk (x), by our hypothesis. Thus F (x) is uniquely determined by x, and we have a single-valued function F : X → Y .Itis clearly an extension of each f j .
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