Qtr. Bryysofx. Bsyly. &B b ,!DD)< If,( r=mff!rs ; g igk yq. p± : }y s. Any compact. disjoint. Det. space. of s. Bsgcy ;) Bry!Yj ),

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2 metric SCE spce Any compct subset S of is closed p± We show tht Sc is S we every yes find disjoint Bryysofx is y s Bsyly n Tke K 5 nbhds For cover of s WTS Brix 5 Qtr of Y &B b compctness of S there exist Y Thm_3 yz S yn finitely mny sit ji U Bsgcy ; then tke rmff!rs ; we hve Brk c c Bry!Yj f Bsy ;D ;!DD< Bsy f g igk yq ; c S 8 Det p metric spce is bounded if s c Bpcxo for some XOE nd > R > 0

3 Borel Let Construct Bs A compct set is bounded p± SC is compct XOE cover U Bnlxo t xo with ndirs Then U is n being compctness there exists mny s t 1 k g U Bnjko j new n blls centered nturl numbers of cover of S s well m nz by Rk finitely Then Let N mxlnj jus ;k Bwko so s is bounded Then S C Thm5[ set is closed nd bounded Generlized Heine ] subsets is compct s is closed & bounded Thm_4 Above two Theorems show tht compct R is NOT necessrily true for generl metric spce To prone the Theorem we use the following def & lemms Det Let z s E bj 2 jl n b bi K in in R Then teed nd b is the set bn ER between stisfying µyjb [ b ] xelr jcxjebj j 3 in Define the dimeter dimfib ] P b n fbjj2

4 Let lemm1_ Let k T be sequence of nested closed intervls tht is closed intervls ts sit Ci 1kt c k KEN ii in km dimko Then µf1k z for some ZER Lenin Any closed intervl in R is compct in R pfotthm Sr be closed nd bounded Then S Brcxo for some Not R & r > 0 ifengie#is*efxieyieiisenbxy4r xitr xii4r Then its plin tht Sc Brcxo C [ b ] [ b ] is compct in R hence b ] itself is compct spce s is closed in [ b ] the Thmz Above we hve S is compct n [ ib ] Compct which implies tht S is in R

5 Bounded Finl Conflict Reminder R n metric generl spce The converse is in generl compctness not true closgdness bldness But in R the converse is true Exmples of tht closednesst boldness p Pc x y infinite set 0 if with discrete metric * y 1 if xty Recll is Clim NOT compct spce y is n blls Bik [ Actully Then U does NOT hve finite U doesnt hve proper ylpcxisk pt Consider the cover U Bik XE cover of with infinitely mny swbcover of since ech bll Bilx sub cover cover one element ] only ie X so none cn be removed in order to cover clim 1 is closed R bold # closed becuse in dol re both closed R becuse Xo PCX 1 fortxo # So C Bzcxo

6 l2 x x x xs / plx EE lxj / lxjl2< FE y ny HE x yet S er o to kf%+ Yjt co for 7 5 ez Clim S is closed R bounded P p er ej p ktj 0 kj S hs no limit point 01W if xoes in nbhd ny of xo points of S there re infinitely mny nbhd of ny point with rdius 1 hs t most 1 pt of s 5 SUS SU S s is closed S is bounded becuse S Bdei Clim S is NOT compct P similr to the previous exmple tke U Bicek F where Beer is the cover # needless bll in l2 to sy is n infinite doesnt swhich hve proper finite sub cover sub cover

7 metric of R This exmple shows tht compctness for closed & bold set is lost when the dimension of the spce becomes A time Equivlent cord compctness spce or generlly topologicl spce is compct spce if nd only if ny R F n fmily of closed subset Fc stisfies tht?f hs finite subfmily F F ; g1 R± The corresponding C R st Sttement forrsc ffj subset being compct # S s subspce with the metric of is compct using reltive topology closed set in S is Fns for some closed set FC P Suppose is compct Given ny R stisfying bove condition t fter 1 ie U FE is n cover of compctness of U hs finite sub cover

8 13 FT g sit FT Fj < so fj 01 suppose with t e ft Fj 1 1 hs the R property H R 01 hs finite subfmily F fj je 4 A Ff closed in 71 Tke ny cover U Ux A of The stisfies fmily of compliments R U ea h U uy P closed the R subfmily F F Us property there exists finite of UJ j of R sit ie G g ZC n j Uuj So we find finite sub cover of for every cover U is compct

9 Every / Rk_ Above proof doesnt use the concept of dist its bsed on topology ie sets / closed sets metric So it generlizes to ny topologicl spce Appliction? Thm_ 1 metric spce only for metric spce The following sttements re equivlent Ci C ii ii is compct Every infinite subset of hs limit point in hs tht sequence subsequence to converges point of in to be R± lmgin bounded R closed #E subset of R G confined! in Pf if ii contrdiction suppose is compct but there exists n infinite subset DC sit D n of Then fort XED since XCTD there exists nbhd Ux of x sit U ndx

10 U Let in tht < let U U D p p cover of D infinite U is n cover pen g of hs no finite infinite removing Ux will remove X coz its sub cover only covered Ux by P is closed since D is compct ii iii xn P be ny sequence of E xn n > is either finite or infinite f E is finite 7 one of the element AEE sit Xn for infinitely mny n s Choose those n s nd order them incresingly we hve swbseq Xnk P where Xnpi f E is infinite ii it hs limit point 1 denoted s Xo EE inductive kl We extrct subseq using the following stndrd step choose n sit k2 Choose nz > n Xn E B Xo sit possible since every hs infinitely mny XnzEB±ko nbhd of Xo Xn s suppose we hve xn nz sxnk with his nz< n < nk RXnkEB Ko we cn choose nk+ > Nk st Xn Bh+TC o Be