Lecture 10: Top ology of the Real Line 10.1 Intervals De nition Given anytwo extended real numb ers a<b, the set a; b=fx : x 2 R;a<x <bg is called an open interval. Given anytwo nite real numb ers a b, the sets [a; b]=fx : x 2 R;a x bg; 1;b]=fx : x 2 R;x bg; and [a; +1=fx : x 2 R;x ag are called closed intervals.Any set whichisanopeninterval, a closed interval, or is given by, for some nite real numb ers a<b, a; b]=fx : x 2 R;a<x bg or [a; b=fx : x 2 R;a x<bg; is called an interval. Prop osition If a; b 2 R, then a; b=fx : x = a +1 b; 0 < < 1g: Pro of Supp ose x = a +1 b for some 0 <<1. Then b x = b a = b a > 0; so x<b, and x a = 1a 1 b =1 b a > 0; so a<x. Hence x 2 a; b. Now supp ose x 2 a; b. Then b x x a b x b x x = a + b = a + 1 b b a b a b a b a and b x 0 < < 1: b a 10-1 Lecture 10: Top ology of the Real Line 10-2 10.2 Op en sets De nition A set U R is said to b e open if for every x 2 U there exists >0 such that x ; x + U: Prop osition Every op en interval I is an op en set. Pro of Supp ose I =a; b where a<b are extended real numb ers. Given x 2 I , let b e the smaller of x a and b x. Supp ose y 2 x ; x + . If b =+1, then b>y; otherwise, wehave b y>b x + =b x b x b x=0; so b>y.Ifa = 1, then a<y; otherwise, y a>x a =x a x a x a=0; so a<y.Thus y 2 I and I is an op en set. Note that R is an op en set it is, in fact, the op en interval 1; +1, as is ; it satis es the de nition trivially. S Prop osition Supp ose A is a set and, for each 2 A, U is an op en set. Then U 2A is an op en set. S U . Then x 2 U for some 2 A. Since U is op en, there exists Pro of Let x 2 2A an >0 such that x ; x + U .Thus [ x ; x + U U : 2A S Hence U is op en. 2A T n Prop osition Supp ose U ;U ;::: ;U is a nite collection of op en sets. Then U is 1 2 n i i=1 op en. T n Pro of Let x 2 U . Then x 2 U for every i =1; 2;:::;n.For each i,cho ose > 0 i i i i=1 such that x ;x + U . Let b e the smallest of ; ;:::; . Then >0 and i i i 1 2 n x ; x + x ;x + U i i i for every i =1; 2;::: ;n.Thus n \ x ; x + U : i i=1 T n Hence U is an op en set. i i=1 Lecture 10: Top ology of the Real Line 10-3 Along with the facts that R and ; are b oth op en sets, the last two prop ositions show that the collection of op en subsets of R form a topology. De nition Let A R.Wesay x 2 A is an interior p ointof A if there exists an >0 such that x ; x + A. The set of all interior p oints of A is called the interior of A, denoted A . Exercise 10.2.1 Show that if A R, then A is op en. Exercise 10.2.2 Show that A is op en if and only if A = A . Exercise 10.2.3 Let U R b e a nonempty op en set. Show that sup U=2 U and inf U=2 U . 10.3 Closed sets De nition A p oint x 2 R is called a limit point of a set A R if for every >0 there exists a 2 A, a 6= x, such that a 2 x ; x + . De nition Supp ose A R. A p oint a 2 A is called an isolatedpoint of A if there exists an >0 such that A \ x ; x + =fag: Exercise 10.3.1 Identify the limit p oints and isolated p oints of the following sets: a [1; 1], b 1; 1, 1 + c : n 2 Z , n d Z, e Q. Exercise 10.3.2 Supp ose x is a limit p oint of the set A. Show that for every >0, the set x ; x + \ A is in nite. 0 We denote the set of limit p oints of a set A by A . 0 De nition Given a set A R, the set A = A [ A is called the closure of A. De nition A set C R is said to b e closed if C = C . Prop osition If A R, then A is closed. Pro of Supp ose x is a limit p ointof A.We will show that x is a limit p ointof A, and hence x 2 A.Now for any >0, there exists a 2 A, a 6= x, such that ;x + : a 2 x 2 2 Lecture 10: Top ology of the Real Line 10-4 If a=2 A, then a is a limit p ointof A, so there exists b 2 A, b 6= a and b 6= x, such that b 2 a ;a + : 2 2 Then jx bjjx aj + ja bj < + = : 2 2 0 A is closed. Hence x 2 A , and so Prop osition A set C R is closed if and only if for every convergent sequence fa g k k 2K with a 2 C for all k 2 K , k lim a 2 C: k k !1 Pro of Supp ose C is closed and fa g is a convergent sequence with a 2 C for all k k 2K k k 2 K . Let x = lim a .Ifx = a for some integer k , then x 2 C . Otherwise, for k !1 k k every >0, there exists an integer N such that ja xj <. Hence a 6= x and N N a 2 x ; x + : N Thus x is a limit p ointof C , and so x 2 C since C is closed. Now supp ose that for every convergent sequence fa g with a 2 C for all k 2 K , k k 2K k lim a 2 C: Let x b e a limit p ointofC .For k =1; 2; 3;:::,cho ose a 2 C such that k !1 k k 1 1 a 2 x ;x + . Then clearly k k k x = lim a ; k k !1 so x 2 C .Thus C is closed. Exercise 10.3.3 Show that every closed interval I is a closed set. T C Prop osition Supp ose A is a set and, for each 2 A, C is a closed set. Then 2A is a closed set. T Pro of Supp ose x is a limit p ointof C . Then for any >0, there exists y 2 2A T C such that y 6= x and y 2 x ; x + . But then for any 2 A, y 2 C ,so x 2A is a limit p ointofC . Since C is closed, it follows that x 2 C for every 2 A.Thus T T x 2 C and C is closed. 2A 2A S n C Prop osition Supp ose C ;C ;:::;C is a nite collection of closed sets. Then i 1 2 n i=1 is closed. S n Pro of Supp ose fa g is a convergent sequence with a 2 C for every k 2 K . k k 2K k i i=1 Let L = lim a . Since K is an in nite set, there must exist an integer m and a k !1 k 1 subsequence fa g such that a 2 C for j =1; 2;:::. Since every subsequence of n n m j j j =1 1 fa g converges to L, fa must converge to L. Since C is closed, g k k 2K n m j j =1 n [ L = lim a 2 C C : n m i j j !1 i=1 Lecture 10: Top ology of the Real Line 10-5 S n C is closed. Thus i i=1 Note that b oth R and ; satisfy the de nition of a closed set. Prop osition A set C R is closed if and only if R n C is op en. Pro of Assume C is closed and let U = R n C .IfC = R, then U = ;, which is op en; if C = ;, then U = R, which is op en. So wemay assume b oth C and U are nonempty. Let x 2 U . Then x is not a limit p ointof C , so there exists an >0 such that x ; x + \ C = ;: Thus x ; x + U; so U is op en. Now supp ose U = R n C is op en. If U = R, then C = ;, which is closed; if U = ;, then C = R, which is closed.