Question

4. Do each of the following: (a) Show that a finite union of compact sets is compact, i.e. given compact sets K1,.., Kn show that K1U .U Kn is compact. (b) Show that an arbitrary intersection of compact sets is compact, i.e. given compact sets {Ka}a where each Ka is compact, show that no Ka is compact. 1

Answer 1

Dost we show that union of two compact sete is Compact Let Ii fkz be two Compact sets . Let k=kiuke Lot ling be sequence ink, Then lanberioka ni Eki (2) poffre - since hand is sequence of Kp for falcone - & Ki is compact, boole has seebsequence enme you maizdas Seuch that same has a convergent Subsequence Suppose of Denme & Correopes to points in the for fl@) the formed is also a seebsequence of angernal sequence fine and kick. fille,

stand has a convergent Sebsequence in a C'exerick, ten Hence every sequence in t has convergent Seebsegence. 1. Af Compact The livion of the Compad sets 13 compad f het Ki, Hz, ... In be guren compact sets - Reet Ka Ukna d love we show that it is compact, by lesing indection on n. If n=2 then of above proof it is compact Assame that the which of an compact sets is Compact Monte ką ks ... An are compact sets, - Kauks u... Ukn is also compact i kako (KEUKSU

Compact since hid keukgu...oko are compact sate by above proof Holka ungu. Uky) is also 1. Hence K= Ok, is compact. (i) Let & Kalle be family of compact sets. Let kan hoc Le show that Ki8 dosed and bacended, since each Kors Compact, each Kassclosed and bounded Since the i blareny intersectors of Caled sels is dosed, Kinken. - Ata is closed - ie n Koc is closed - OCEA" nic It is closed since each Koc is bounded, HiQG₂n-.ntorno... 1 is also bounded 2 pc & Deser KAS bocinded Hence it is closed and bounded

15 kW Compac © The union of infinitely marry compact sets is not compact For Bample we know that in is not compact because the sequence of naboezel reembers amb has no convessent seeb sequence CO) R 18 not bounded. CARL10,00) because th= y Eni na where each internal [min] is closed of bounded and hence Compact Joter we prove @ 45 by using properties of his compact KM Compact <) every sequence of it has convergent Seebsequence Al compact c) A is closed & bounded