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Pragmatik Anˆlush (2010 11) TopologÐa metrik n q rwn Ask seic Omˆda A' 1. 'Estw (X, ρ) metrikìc q roc kai F, G uposônola tou X. An to F eðnai kleistì kai to G eðnai anoiktì, deðxte ìti to F \ G eðnai kleistì kai to G \ F eðnai anoiktì. 2. 'Estw (X, ρ) metrikìc q roc. DeÐxte ìti kˆje uposônolo A tou X grˆfetai wc tom anoikt n uposunìlwn tou (X, ρ). 3. 'Estw f : R R suneq c sunˆrthsh. DeÐxte ìti to G = {x R : f(x) > 0} eðnai anoiktì uposônolo tou R kai to F = {x R : f(x) = 0} eðnai kleistì uposônolo tou R. 4. DeÐxte ìti kˆje kleistì diˆsthma sto R grˆfetai wc arijm simh tom anoikt n diasthmˆtwn kai kˆje anoiktì diˆsthma sto R grˆfetai wc arijm simh ènwsh kleist n diasthmˆtwn. 5. ApodeÐxte ìti kˆje peperasmèno uposônolo enìc metrikoô q rou eðnai kleistì. 6. ApodeÐxte ìti kˆje sfaðra enìc metrikoô q rou eðnai kleistì sônolo. MporeÐ se ènan metrikì q ro mia sfaðra na eðnai to kenì sônolo? 7. 'Estw (X, d) metrikìc q roc, x X kai ε > 0. Exetˆste, an isqôei pˆntote h isìthta B(x, ε) = {y X : d(x, y) ε}. [UpenjÔmish: Gia kˆje A X sumbolðzoume me A thn kleist j kh tou A.] 8. 'Estw (X, d) metrikìc q roc. H diag nioc tou X X eðnai to sônolo = {(x, x) : x X}. ApodeÐxte ìti to eðnai kleistì ston X X wc proc th metrik d 2, ìpou d 2 ((x 1, y 1 ), (x 2, y 2 )) = d 2 (x 1, y 1 ) + d 2 (x 2, y 2 ). Genikìtera, apodeðxte ìti to eðnai kleistì wc proc kˆje metrik ginìmeno ston X X. 9. Upˆrqei ˆpeiro kleistì uposônolo tou R to opoðo apoteleðtai mìno apì rhtoôc? Upˆrqei anoiktì uposônolo tou R to opoðo apoteleðtai mìno apì ˆrrhtouc? 10. 'Estw A, B dôo uposônola enìc metrikoô q rou (X, d). ApodeÐxte ìti: (a) An A B = X, tìte A B = X. (b) An A B =, tìte A B =. 11. 'Estw (X, d) metrikìc q roc. ApodeÐxte ìti: (a) (A \ B) A \ B gia kˆje A, B X. (b) A \ B A \ B gia kˆje A, B X. MporoÔme na antikatast soume touc egkleismoôc me isìthtec? 12. 'Estw (X, ρ) metrikìc q roc kai = A X. DeÐxte ìti diam(a) = diam(a). IsqÔei to Ðdio gia to eswterikì tou A? 13. (a) 'Estw A anoiktì uposônolo tou (X, ρ) kai G A. DeÐxte ìti to G eðnai anoiktì sto A an kai mìno an eðnai anoiktì ston X. (b) 'Estw A kleistì uposônolo tou (X, ρ) kai G A. EÐnai swstì ìti to G eðnai kleistì sto A an kai mìno an eðnai kleistì ston X? 14. BreÐte èna arijm simo kai puknì uposônolo tou R \ Q wc proc th sun jh metrik.

Omˆda B' 15. 'Estw (X, ) q roc me nìrma. DeÐxte ìti B(x, r) = B(x, r) gia kˆje x X kai kˆje r > 0. 16. DeÐxte ìti o c 0 eðnai kleistì uposônolo tou l. Ti mporeðte na peðte gia ton c 00? EÐnai anoiktì uposônolo tou l? kleistì uposônolo tou l? 17. 'Estw (X, ρ) metrikìc q roc. DeÐxte ìti ta akìlouja eðnai isodônama: (a) To G eðnai anoiktì. (b) Gia kˆje A X, G A G A. (g) Gia kˆje A X, G A = G A. 18. DeÐxte ìti kˆje anoiktì uposônolo tou R grˆfetai wc ènwsh arijm simwn to pl joc anoikt n diasthmˆtwn me rhtˆ ˆkra. 19. ApodeÐxte ìti sto R den upˆrqoun mh tetrimmèna uposônola (dhlˆd diaforetikˆ apì to kai to R) ta opoða na eðnai sugqrìnwc anoiktˆ kai kleistˆ. 20. (a) Gia kˆje n Z, èstw F n kleistì uposônolo tou (n, n + 1). Jètoume F = n Z F n. ApodeÐxte ìti to F eðnai kleistì sto R. Upìdeixh: DeÐxte pr ta ìti gia kˆje n upˆrqei δ n > 0 ètsi ste x y δ n opoted pote x F n kai y F m, n m. (b) BreÐte mia akoloujða xènwn anˆ duo kleist n sunìlwn sto R twn opoðwn h ènwsh den eðnai kleistì sônolo. 21. 'Estw (X, d) metrikìc q roc. ApodeÐxte ìti: (a) An to X èqei perissìtera apì èna stoiqeða, tìte upˆrqei anoiktì G X, ste G kai X \ G. (b) An to X eðnai ˆpeiro sônolo, tìte upˆrqei anoiktì G X ste to G kai to X \ G na eðnai ˆpeira. 22. Estw (X, ρ) metrikìc q roc kai x, y X me x y. DeÐxte ìti upˆrqoun anoiktˆ sônola U, V ste x U, y V kai U V =. 23. Estw (X, ρ) metrikìc q roc, x X kai F kleistì uposônolo tou X me x / F. DeÐxte ìti upˆrqoun anoiktˆ sônola U, V ste x U, F V kai U V =. MporoÔme na petôqoume na isqôei, epiplèon, ìti U V =? 24. 'Estw (X, ρ) metrikìc q roc kai A X. Jètoume A to parˆgwgo sônolo tou A, dhlad to sônolo twn shmeðwn susss reushc tou A. ApodeÐxte ta akìlouja: (a) A = A A. Sumperˆnate ìti to A eðnai kleistì an kai mìno an perièqei ta shmeða suss reus c tou. (b) To A eðnai kleistì sônolo. (g) An A B X tìte A B. (d) A = (A). Dhlad, ta A kai A èqoun ta Ðdia shmeða suss reushc. (e) (A ) A. BreÐte uposônolo A tou R ste o egkleismìc na eðnai gn sioc. 25. Exetˆste an oi akìloujoi isqurismoð eðnai alhjeðc: (a) Upˆrqei A R ste A = N. (b) Upˆrqei A R ste A = Z. (g) Upˆrqei A R ste A = Q. 26. 'Estw (X, ρ) metrikìc q roc. An A, B X, h apìstash tou A apì to B orðzetai wc ex c: dist(a, B) = inf{ρ(a, b) : a A, b B}. ApodeÐxte tic akìloujec idiìthtec thc apìstashc:

(a) an A B, tìte dist(a, B) = 0. (b) dist(a, B) = dist(a, B). (g) dist(a, B C) = min{dist(a, B), dist(a, C)}. (d) D ste parˆdeigma kleist n kai xènwn uposunìlwn A, B enìc metrikoô q rou (X, ρ) ta opoða èqoun mhdenik apìstash. 27. 'Estw (X, ρ) metrikìc q roc kai A X. An x X orðzoume thn apìstash tou x apì to A na eðnai h apìstash twn sunìlwn {x}, A: dist(x, A) = inf{ρ(x, a) : a A}. ApodeÐxte ìti: (a) dist(x, A) = 0 an kai mìno an x A. (b) dist(x, A) dist(y, A) ρ(x, y) gia kˆje x, y X. (g) To sônolo {x X : dist(x, A) < ε} eðnai anoiktì, en to sônolo {x X : dist(x, A) ε} eðnai kleistì. (d) An A B A, tìte dist(x, A) = dist(x, B) gia kˆje x X. 28. 'Estw (X, ρ) metrikìc q roc kai A X. ApodeÐxte ìti A = {x X : dist(x, A \ {x}) = 0}. 29. 'Estw (X, ρ) metrikìc q roc. ApodeÐxte ìti kˆje kleistì uposônolo tou X grˆfetai wc arijm simh tom anoikt n sunìlwn kai kˆje anoiktì uposônolo tou X grˆfetai wc arijm simh ènwsh kleist n sunìlwn. 30. 'Estw (X, ρ) metrikìc q roc kai A X. ApodeÐxte tic ex c idiìthtec tou sunìrou tou A: (a) bd(a) = bd(a c ). (b) cl(a) = bd(a) A. (g) X = A bd(a) (X \ A). (d) bd(a) = A \ A isodônama bd(a) = A X \ A. Epomènwc, to sônoro eðnai kleistì sônolo. (e) To A eðnai kleistì an kai mìno an bd(a) A. 31. 'Estw (X, ρ) metrikìc q roc kai A, B X. ApodeÐxte ta akìlouja: (a) An to A eðnai anoiktì kleistì uposônolo tou X tìte to bd(a) èqei kenì eswterikì. (b) An A B = tìte bd(a B) = bd(a) bd(b). 32. BreÐte uposônolo A tou R ste (bd(a)) = R. 33. 'Estw A uposônolo tou (X, ρ). An G kai H eðnai xèna anoiktˆ sônola sto A, deðxte ìti upˆrqoun xèna anoiktˆ sônola U kai V sto X ste G = A U kai H = A V. 34. 'Estw (X, ρ) diaqwrðsimoc metrikìc q roc. DeÐxte ìti kˆje oikogèneia xènwn anoikt n uposunìlwn tou X eðnai peperasmènh arijm simh. 35. 'Estw (X, ρ) metrikìc q roc. DeÐxte ìti: (a) An D eðnai èna puknì uposônolo tou X, tìte D G = G gia kˆje anoiktì uposônolo G tou X. (b) An to G eðnai anoiktì kai puknì uposônolo tou X kai to D eðnai puknì uposônolo tou X, tìte to G D eðnai puknì uposônolo tou X. IsqÔei to Ðdio an to G den upotejeð anoiktì? (g) EÐnai swstì ìti h tom miac akoloujðac anoikt n kai pukn n uposunìlwn tou X eðnai puknì uposônolo tou X? 36. 'Estw (X 1, d 1 ),..., (X n, d n ) metrikoð q roi. JewroÔme ton q ro ginìmeno (X, d) me X = n i=1 X i kai d = max 1 i n d i. DeÐxte ìti:

(a) An kˆje G i eðnai d i -anoiktì ston X i, i = 1,..., n, tìte to n i=1 G i eðnai d-anoiktì ston X. (b) An kˆje F i eðnai d i -kleistì ston X i, i = 1,..., n, tìte to n i=1 F i eðnai d-kleistì ston X. (g) An kˆje D i eðnai puknì ston X i, i = 1,..., n, tìte to D = n i=1 D i eðnai puknì ston X. Eidikìtera, an kˆje (X i, d i ), i = 1,..., n eðnai diaqwrðsimoc tìte o (X, d) eðnai diaqwrðsimoc. Omˆda G' 37. 'Estw (X, ρ) metrikìc q roc kai P X. To P lègetai tèleio an eðnai kenì eðnai kleistì kai kˆje shmeðo tou eðnai shmeðo suss reushc gi autì. ApodeÐxte ta akìlouja: (a) 'Ena sunìlo P (X, ρ) eðnai tèleio an kai mìno an P = P. (b) Kˆje kleistì (mh tetrimmèno) diˆsthma sto R (me th sun jh metrik ) eðnai tèleio sônolo. EpÐshc, to R eðnai tèleio an jewrhjeð wc uposônolo tou R 2. (g) Kˆje mh kenì tèleio uposônolo P tou R eðnai uperarijm simo. [Upìdeixh. To P eðnai ˆpeiro. An eðnai arijm simo, grˆfetai sth morf P = {x n : n N}. OrÐste katˆllhlh akoloujða kibwtismènwn diasthmˆtwn [a n, b n ] ste, gia kˆje n N, [a n, b n ] P allˆ x n / [a n, b n ].] 38. 'Estw A R kai x R. To x lègetai shmeðo sumpôknwshc tou A an gia kˆje ε > 0 to sônolo A (x ε, x + ε) eðnai uperarijm simo. ApodeÐxte ta akìlouja: (a) An to A eðnai arijm simo tìte den èqei shmeða sumpôknwshc. (b) An to A eðnai uperarijm simo kai P eðnai to sônolo twn shmeðwn sumpôknwshc tou A tìte P = P kai to A \ P eðnai arijm simo. (g) An to A eðnai kleistì uposônolo tou R tìte upˆrqoun tèleio sônolo P kai arijm simo sônolo Z ste A = P Z kai P Z =. 39. 'Estw (X, ρ) metrikìc q roc kai (x n ) akoloujða sto X. To x X lègetai oriakì shmeðo ρ thc (x n ) an upˆrqei upakoloujða (x kn ) thc (x n ) ste x kn x. Jètoume L(x n ) to sônolo twn oriak n shmeðwn thc akoloujðac (x n ). ApodeÐxte ìti ρ (a) An x n x tìte L(x n ) = {x}. IsqÔei to antðstrofo? (b) An A = {x n : n N} X tìte A L(x n ) A. DeÐxte me èna parˆdeigma ìti oi egkleismoð mporeð na eðnai gn sioi. (g) DeÐxte ìti to L(x n ) eðnai kleistì uposônolo tou X. (d) An to A den eðnai kleistì, deðxte ìti L(x n ). An epiplèon, h (x n ) eðnai ρ-cauchy, tìte eðnai ρ-sugklðnousa. (e) To x eðnai oriakì shmeðo thc (x n ) an kai mìno gia kˆje ε > 0 kai gia kˆje n N upˆrqei m n ste x m B ρ (x, ε). 40. Swstì lˆjoc? Gia kˆje ˆpeiro metrikì q ro (X, d) upˆrqei ˆpeiro uposônolo A tou X ste kˆje G A na eðnai anoiktì wc proc th sqetik metrik sto A. 41. 'Estw (X, ρ) diaqwrðsimoc metrikìc q roc. ApodeÐxte ìti: (a) To sônolo twn memonwmènwn shmeðwn tou X eðnai to polô arijm simo. (b) An S eðnai èna uperarijm simo uposônolo tou X tìte upˆrqei akoloujða diaforetik n anˆ duo stoiqeðwn tou S, h opoða sugklðnei se shmeðo tou S. 42. 'Estw θ R \ Q. DeÐxte ìti to sônolo D(θ) := {(cos(2πnθ), sin(2πnθ)) : n N} eðnai puknì ston kôklo S 1 = {(x, y) R 2 : x 2 + y 2 = 1}. 43. 'Estw (X, ρ) metrikìc q roc. To A X lègetai poujenˆ puknì an int(a) =. ApodeÐxte ìti: (a) To A X eðnai poujenˆ puknì an kai mìnon an A (X \ A). (b) To A X eðnai poujenˆ puknì kai kleistì an kai mìnon an to X \ A eðnai puknì kai anoiktì.

(g) An to A eðnai kleistì uposônolo tou X, tìte to A eðnai poujenˆ puknì an kai mìnon an A = bd(a). (d) An to A eðnai poujenˆ puknì uposônolo tou X kai to X \ B eðnai puknì tìte to X \ (A B) eðnai puknì ston X. (e) H ènwsh peperasmènou pl jouc poujenˆ pukn n uposunìlwn tou X eðnai poujenˆ puknì uposônolo tou X. 44. 'Estw (q n ) mia arðjmhsh tou Q. OrÐzoume ( I n = q n 1 2 n, q n + 1 ) 2 n, n N. DeÐxte ìti to U = n=1 I n eðnai anoiktì kai puknì uposônolo tou R kai ìti to U c eðnai poujenˆ puknì. 45. 'Estw (X, ρ) metrikìc q roc kai A X. ApodeÐxte ìti ta akìlouja eðnai isodônama: (a) To A eðnai poujenˆ puknì. (b) To A den perièqei mh kenì anoiktì sônolo. (g) Kˆje mh kenì anoiktì uposônolo tou X perièqei èna mh kenì anoiktì sônolo xèno proc to A. (d) Kˆje mh kenì anoiktì uposônolo tou X perièqei mia anoikt mpˆla xènh proc to A. 46. 'Estw (X n, ρ n ), n = 1, 2,... akoloujða metrik n q rwn me ρ n (x, y) 1 gia kˆje x, y X n, n = 1, 2,.... JewroÔme to q ro ginìmeno (X, ρ), ìpou X = n=1 X n kai ρ(x, y) = n=1 2 n ρ n (x(n), y(n)). StajeropoioÔme α = (α(n)) ston X. JewroÔme ta sônola kai orðzoume D m = {x = (x(n)) X : x(n) = α(n), n > m}, m = 1, 2,... ApodeÐxte ìti to D α eðnai puknì ston X. D α := m=1 47. 'Estw A, B arijm sima, puknˆ uposônola tou R. DeÐxte ìti upˆrqei sunˆrthsh f : A B h opoða eðnai aôxousa, 1-1 kai epð. D m.