MA651 Topology. Lecture 1. Elements of Set Theory 1.This text is based on the book ”Topology” by James DugundgjiI have intentionally made several mistakes in this text. The first homework assignment is to findthem.1 Basic NotationSymbolic logic notation.Definition 1.1. If p and q are propositions, then:p ∨ q (read: p or q) denotes the dis junction of p and q. The assertion ”p ∨ q” is true whenever atleast one of p,q is true.p ∧ q (read: p and q) denotes the conjunction of p and q. The assertion ”p ∧ q” is true only incase both p and q are true.¬ q (read: not q) denotes the negation of p. The assertion ”¬ q” is true only if q is false.p ⇒ q is read: p implies q. By definition, ”p ⇒ q” denotes ”(¬p) ∨ q”. In particular, thestatement ”p ⇒ q” is true if and only if the statement ”(¬ q) ⇒ (¬ p)” is true.p ⇔ q is read: p is logically equivalent to q. By definition, ”p ⇔ q” denotes ”(p ⇒ q) ∧ (q ⇒ p)”.An expression p (x) that becomes a proposition whenever values from a specified domain of dis-course are substituted for x is called a propositional function or, equivalently, a condition on x;and p is called a property, or predicate. The assertion ”y has property p” means that ”p(y)” istrue. Thus, if p(x) is the proportional function ”x is an integer”, then p denotes the property ”isan integer”, and ”p(2)” is true whereas ”p(1/2)” is false.The quantifier ”there exists” is denoted by ∃, an the quantifier ”for each” is denoted by ∀. Theassertion ”∀x ∃y ∀z : p(x, y, z)” reads ”For each x there exists a y such that for each z, p(x, y, z)is true”; its negation is obtained mechanically by changing the sense of each quantifier (preservingthe given order of the variables!) and negating the proposition: thus, ”∃x ∀y ∃x : ¬ p(x, y, z)”.12 SetsIntuitively, we think of a set as something made up by all the objects that satisfy some givenconditions, such as the set of prime numbers, the set of points on a line, or the set of objectsnamed in a given list. The objects making up the set are called the elements, or members, of theset and may themselves be sets, as in the set of all lines in the plane.Rigorously, the word set is an undefined term in mathematics, so that definite axioms are requiredto govern the use of this term. Although we shall deal with sets on an intuitive basis until wediscuss an axiom systems, whenever we apply the label set to something, we shall later providethis usage to have been formally justified.The membership relation is denoted by ∈ and sets are generally indicated by capital letters:”a ∈ A” is read ”a belongs to (is a member, eleme nt, point of) the set A”; ¬(a ∈ A) is writtena 6∈ A. The notation a = b will mean that the objects a and b are the same, and a 6= b denotes¬(a = b). If A, B sets, then A = B will indicate that A and B have the same elements; that is,∀x : (x ∈ A) ⇔ (x ∈ B); ¬(A = B) is written A 6= B.A ⊂ B (or B ⊃ A), read ”A is a subset of (is contained in) B”, signifies that each element of Ais an element of B, that is ∀x : (x ∈ A) ⇒ (x ∈ B); e quality is not excluded - we call A a propersubset of B (A ( B whenever (A ⊂ B) ∧ (A 6= B). The relations ⊂ and ( are called inclusionand proper inclusion, respectively. the following statements are evident:Proposition 2.1. A ⊂ A for each set A.Proposition 2.2. If A ⊂ B and B ⊂ C, then A ⊂ C (that is, C is transitive).Proposition 2.3. A = B if and only if both A ⊂ B and B ⊂ A.Of these, the last statement is very important: the equality of two sets is usually proved by show-ing each of the two inclusions valid.The axioms of set theory allow only two methods for forming subsets of a given set. One of theseis by appeal to the axiom of choice, and will be discussed later. The other is by use of the followingprinciple: If A is a set and p is a property that each element of A either has or does not have, thenall the x ∈ A having the property p from a set. This subset of A is denoted by {x ∈ A | p(x)}; itis uniquely determined by the property p. Clearly, {x ∈ A | p(x)} ⊂ {x ∈ A | q(x)} if and only if∀x ∈ A : p(x) ⇒ q(x); thus two properties determine the same subset of A whenever each objectin A having one of them also has the other.Example 2.1. If A is the real line, the closed unit interval is {x ∈ A | 0 6 x 6 1}2Example 2.2. If A is the real line, {x ∈ A | x2= 1} = {x ∈ A | x4= 1} even thought the definingproperties are different. Note that if A was the set of complex numbers, these two properties wouldnot determine the same subset.Example 2.3. For each set A, {x ∈ A | x = x} = {x ∈ A | x ∈ A} = AFor each set A, the null subset ØA⊂ A is {x ∈ A | x 6= x}; it has no members, since each x ∈ Asatisfies x = x.Proposition 2.4. All null subsets are equal. Thus there is one and only one null set, Ø, and itis contained in every set: Ø ⊂ A for every set AProof. Let A, B be any two sets. If ØA⊂ ØBwere false, these would be at least one elementa in ØAnot in ØB; in particular, we would then have as an a ∈ A such that a 6= a and thisis impossible. In the same way. ØB⊂ ØA; therefore, by (2.3), ØA= ØBand all null sets areequal.3 Boolean AlgebraDefinition 3.1. Let Γ be a given s et, and A, B two subsets. The union, ASB, of A and B is{x ∈ Γ | x belongs to at least one of A, B}. The intersection, ATB, of A and B is {x ∈ Γ | xbelongs to both A and B}.According to the definition, a necessary and sufficient condition for two sets A, B to have elementsin common is that ATB 6= Ø; if ATB = Ø, the sets A and B are called disjoint. The followingtwo statements are immediate consequences of (3.1):Proposition 3.1. For any two sets A, B, always ATB ⊂ A ⊂ ASBProposition 3.2. If A ⊂ C and B ⊂ D, then ASB ⊂ CSD and ATB ⊂ CTDThe formal properties of the operationsSandTare given inTheorem 3.1. Each of the operationsSandTis1. Idempotent: ∀A : ASA = A = ATA.2. Associative: AS(BSC) = (ASB)SC and AT(BTC) = (ATB)TC3. Commutative: ASB = BSA and ATB = BTA3Furthermore,Tdistributes overSandSdistributes overT:A\(B[C) = (A\B)[(A\C),A[(B\C) = (A[B)\(A[C).Proof. Verification of (1) − (3) is trivial. To give an …