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**Extra info for Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2)**

**Sample text**

Then y E V, which implies that there exists an x E E such that y E B(X,d)(x,rx)· But since y is also in Y, this implies that Y E B(Y,diYxY)(x, rx)· But by definition of rx, this means that y E E, as desired. Thus we have found an open set V for which E = V n Y as desired. 408 12. Metric spaces Now we do the converse. Suppose that E = V n Y for some open set V; we have to show that E is relatively open with respect to Y. Let x be any point in E; we have to show that x is an interior point of E in the metric space (Y,diYxY)· Since x E E, we know x E V.

7. 1. 7 is indeed a metric space. 8. 1). (For the first inequality, square both sides. 5). 9. 9 is indeed a metric space. 10. 2). 11. 11 is indeed a metric space. 12. 18. 13. 19. 14. 20. 15. Let 00 X:= {(an):=o: L lanl < oo} n=O be the space of absolutely convergent sequences. Define the l 1 and metrics on this space by 00 dtl((an):=O• (bn):=o) := L n=O ian- bnl; zoo 400 12. Metric spaces Show that these are both metrics on X, but show that there exist sequences x< 1>, x< 2 >, ... , sequences of sequences) which are convergent with respect to the d1oo metric but not with respect to the d11 metric.

0 12. 5. Compact metric spaces 417 We close this section by listing some miscellaneous properties of compact sets. 10. Let (X, d) be a metric space. (a) lfY is a compact subset of X, and Z if and only if Z is closed. ~ Y, then Z is compact (b) If Y1, ... , Yn are a finite collection of compact subsets of X, then their union Y1 U ... U Yn is also compact. (c) Every finite subset of X (including the empty set) is compact. Proof. 7. 1. 3 match when talking about subsets of the real line with the standard metric.