Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics)

We have now hence proven finite subcollection of { Va } covers F. Corollary If F is closed and okay is compact, then Fn okay is compact. evidence Theorems 2. 24(b) and a couple of. 34 convey that F n okay is closed ; stnce F n okay c ok, Theorem 2. 35 exhibits that F n ok is compact. 2. 36 Theorem /f{ Ka} is a suite of compact subsets of a metric house X such that the intersection of each finite subcollection of {Ka} is nonempty, then n Ka is nonempty. facts repair a member K1 of { Ka} and positioned Ga = okay� . think that no aspect of K1 belongs to each Ka . Then the units Ga shape an open hide of K1 ; , an such and si nce K1 is compact, there are finitely many indices ct 1 , that K1 c Ga 1 u · · · u Ga , . yet which means • K1 n Ka , n · · · n • • Ka, is empty, in contradiction to our speculation. Corollary If {Kn} is a series of nonempty compact units such that Kn (n = I , 2, three, . . . ), then n f Kn isn't really empty. 2. 37 Theorem element in okay. :::> Kn + t If E is an infin ite subset of a compact set okay, then E has a restrict facts If no element of ok have been a restrict aspect of E, then every one q E okay could have a neigh borhood Vq which includes at so much one aspect of E (namely. q , if q E £). it really is transparent that no finite subcollection of { Vq} can disguise E ; and an analogous is right o f okay, given that E c okay. This contradicts the com pactness of ok. 2. 38 Theorem If {In } is a seq uence of periods in R 1 , such that during (n = 1 , 2, three, . . . ), then nf In isn't really empty. :::> In + 1 facts If In = [an , bn ] permit E be the set of all an . Then E is nonempty and bounded above (by b 1 ) enable x be the sup of E. If n1 and n are positi ve i ntegers, then , . in order that x < b'" for every m. Si nce i t is apparent that am < x E I for m = I , 2, three, .. m . . x, we see that BASIC TOPOLOGY 39 2. 39 'fheorem permit ok be a good integer. If {In} is a series of k-cel/s such that during ::) In + 1 (n = I , 2, three , . . . ) , then n f In isn't empty. facts permit In encompass all poi nts x = (x1, • • • (I < j < okay ; n , xk) such that = 1, 2, three, . . . ) , and installed . i = [an ,j , bn ,j1 · for every j, the series {/n,j} satisfies the hypotheses of Theorem 2. 38. for that reason there are genuine n umbers x;( I < j < okay) such that ( I ; In isn't coated via any finite subcollection of {G«} ; if X E In and Y E In , then I X y I < 2 - n b.