Chapter 2. Basic Topology. 2.1 Finite Countable and Uncountable Sets

Chapter 2. Basic Topology.

2.1 Finite Countable and Uncountable Sets. (continued)

2.6 Remark A ﬁnite set cannot be equivalent to one of its proper subsets. This is possible, however, for inﬁnite sets. What’s an exam- ple??

2.7 Deﬁnition By a sequence, we mean a function f deﬁned on a set J of all positive integers. If f(n)=xn,forn ∈ J, it is custom- ary to denote the sequence f by the symbol {xn}, or sometimes by x1,x2,x3,.... The val- ues of f, that is, the elements xn, are called the terms of the sequence. If A is a set and if xn ∈ A for all n ∈ J, then {xn} is said to be a sequence in A,orasequence of elements of A.

1 2.8 Theorem Every inﬁnite subset of a countable set A is countable.

2.9 Deﬁnition Let A and Ω be sets, and suppose that with each element α of A there is associated a subset of Ω which we denote by Eα. The union of the sets Eα is deﬁned to be the set S such that x ∈ S if and only if x ∈ Eα for at least one α ∈ A. We use the notation

S = ∪α∈AEα. If A consists of the integers 1, 2,...,n, one usu- ally writes n S = ∪m=1Em. If A is the set of all positive integers, the usual notation is ∞ S = ∪m=1Em. Here, ∞ indicates that the union of a countable collection of sets is taken.

2 The intersection of the sets Eα is deﬁned to be the set P such that x ∈ P if and only if x ∈ Eα for every α ∈ A. We use the notation

S = ∩α∈AEα. If A consists of the integers 1, 2,...,n, one usu- ally writes n S = ∩m=1Em. If A is the set of all positive integers, the usual notation is ∞ S = ∩m=1Em. If A ∩ B =6 ∅, we say that A and B intersect; otherwise, they are disjoint.

3 The commutative, associate, and distributive laws hold for unions and intersections:

A ∪ B = B ∪ A;

A ∩ B = B ∩ A;

(A ∪ B) ∪ C = A ∪ (B ∪ C) ;

(A ∩ B) ∩ C = A ∩ (B ∩ C) ; and

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) . Note also that

A ∪∅= A;

A ∩∅= ∅; and if A ⊂ B then

A ∪ B = B and A ∩ B = A.

4 2.12 Theorem Let {En},n =1, 2, 3,...,bea sequence of countable sets, and put ∞ S = ∪n=1En. Then S is countable.

Corollary Suppose A is at most countable, and, for every α ∈ A, Bα is at most countable. Put

T = ∪α∈ABα. Then T is at most countable.

2.13 Theorem Let A be a countable set, and let Bn be the set of all n-tuples (a1,...,an), where ak ∈ A(k =1,...,n), and the elements a1,...,an need not be distinct. Then Bn is countable.

Corollary The set of all rational numbers is countable.

2.14 Theorem Let A be the set of all se- quences whose elements are the digits 0 and 1. This set is uncountable. 5 Say whether each of the following statements is true or false. (a) If Y is a set and {Gα|α ∈ A} is a family of subsets of Y , then ∪α∈AGα is a subset of Y . (b) The range of any sequence is at most countable. (c) The union of any collection of countable sets is countable. (d) If A and B are countable sets, then A ∪ B is countable. (e) If A is countable and B is any set, then A ∩ B is countable. (f) If A is countable and B is any set, then A ∩ B is at most countable.

6 An inﬁnite image of a countable set is countable.

Proof