Question

Exercise 4. [10 marks For every n EN, we define the union and intersection of a collection of n sets A1, A2, ..., An as n UAk = {2 : 3k € {1,...,n} Te Ak} and Ag = {2: Vk € {1,..., n} TE Ax}. k=1 k=1 We define the union and intersection of an infinite collection of sets A1, A2, ...,Ak,... as Ū4x = {2: JkEN 7€ Ax} and ņ 4x = {#: V EN 1 € Ag}. k=1 k=1 (1) Show that if A1, A2, ..., Aks ... is an infinite collection of open sets, then Um, Ak is open. Show that for every n EN, if A1, A2, ..., An is a finite collection of open sets, then nk=1 Ak is open. Provide a concrete example of an infinite collection of open intervals 11, 12, ...,Ik,... such that n_ Ik is not open. Hint: You may for example look for a collection of open intervals of the form (a1, 1), (a2, 1), ..., (ak, 1), ... with numbers 21, 22, ..., Ak, ... cho- sen in such a way that nmi (ak, 1) = (0,1).

Answer 1

A Date..... 1. To show that A, A2, -, Ak is an Infinite collection of open sets, then & A4 K4 Is open Proof cet (xad) be a Metric space ut I & AKYKEN be any arbitrary collection of opens sets of x - let H = U AK case i. If H=0 then His open . empty Set close and open both. C4 1. 1 H+ Then to show t is open TO snow: 7 8yo such that open ball S (22) CH let a ¢ H = Q AKT a AS a E Am for some MEN a tam is open Set 7 870 such that open ball S (28) Cam H. S (2,8) T U AK K 1

Date... ů Ak is open Abrtvary union of open set is open set. show that If A A2 - An, NEN is a finite collection of 1 open open sets, t sets, then nAr to open Proof! Let (x, al be a metric space. let & Ax}x= 1, 2, n be any fiulte collection of open sets. Let A = ñ Ai open To show : A is open set case i of AF & then A is case it. If A to To show! A is open to show! 7 870 such that S(XV) SA AE A = Ak open ball n at n Ar

Date - at Ak k=1,2,.. For each K=1, 2, n. ata s open . 7 8. To Such that open ball Sair w g min S x K=1,2,-, n} > 0 then r r & k = 1,2; ., n. Sar) & S(2.8k) c AR & K=1, 2, - S(28) S = A in S(x,x) S A . Ae . n Ax is open finite Intersection of open set to open rii. let in CR, usual metale) et AK = (- / 1 / 3 & A = (-1, 1) A = (- 12) A = (-1/2 1 4 ) El

Date.. there each Ax is open But NAK oh which not open So? interval Reason! let of possible it is open then I rro such that co-8,0+8) * Gr. r s oz which is an absurd as every is Infinite {o} is not open Infinite Intersection of open need not be open. sets