Genik TopologÐa kai Efarmogèc

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1 Genik TopologÐa kai Efarmogèc

2 ii

3 Perieqìmena iii

4 iv PERIEQŸOMENA

5 Kefˆlaio 1 TopologikoÐ q roi 1.1 TopologÐa Orismìc 1.1. 'Estw X mh kenì sônolo kai T mða oikogèneia uposunìlwn tou X. H T kaleðtai topologða tou X, an ikanopoieð tic parakˆtw idiìthtec: (i) To X kai to an koun sthn T. (ii) H T eðnai kleist stic peperasmènec tomèc, dhlad, gia kˆje n N kai {U i } n i=1 peperasmènh oikogèneia stoiqeðwn thc T, isqôei n i=1 U i T. (iii) H T eðnai kleist stic aujaðretec en seic, dhlad, gia kˆje sônolo I (pou lìgw thc (i), arkeð na eðnai diˆforo tou kenoô) kai {U i } i I oikogèneia stoiqeðwn thc T, isqôei i I U i T. To sônolo X efodiasmèno me mða topologða T kaleðtai topologikìc q roc kai sumbolðzetai me to zeôgoc (X, T ). Ta stoiqeða thc topologðac T kaloôntai anoiktˆ uposônola tou topologikoô q rou (X, T ). UpenjÔmish 1.2. 'Estw (X, ρ) metrikìc q roc kai A X. (i) To A kaleðtai anoiktì uposônolo tou metrikoô q rou X, an gia kˆje x A, upˆrqei ε > 0 tètoio, ste B(x, ε) A. (ii) To A kaleðtai kleistì uposônolo tou metrikoô q rou X, an to X A eðnai anoiktì. 1

6 2 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI Prìtash 1.3. 'Estw (X, ρ) m.q. kai T ρ h oikogèneia twn anoikt n uposunìlwn tou X. H T ρ apoteleð mia topologða tou X kai onomˆzetai epagìmenh topologða tou X apì th metrik ρ, metrik topologða tou X ìtan h metrik eðnai gnwst. Apìdeixh. Ja deðxoume ìti h T ρ ikanopoieð tic treic idiìthtec tou orismoô thc topologðac. (i) Profan c X X. EpÐshc, gia kˆje x X kai gia kˆje ε > 0 (ˆra, sðgoura upˆrqei èna ε > 0), prokôptei ex orismoô tou sunìlou thc mpˆlac B(x, ε) := {y X : ρ(x, y) < ε} ìti B(x, ε) X. Epomènwc, X T ρ. To eðnai epðshc anoiktì uposônolou tou metrikoô q rou X. Prˆgmati, èstw ìti X. Tìte, ja èprepe na upˆrqei x, ste x X, to opoðo eðnai ˆtopo. OmoÐwc, epalhjeôoume ìti eðnai anoiktì. Epomènwc, T ρ. (ii) 'Estw n N kai {U i } n i=1 peperasmènh oikogèneia anoikt n uposunìlwn tou X. An n i=1 U i =, tìte n i=1 U i T ρ. 'Estw x n i=1 U i. Tìte, gia kˆje i {1,..., n}, upˆrqei ε i > 0 tètoio, ste B(x, ε i ) U i. Jètoume ε = min i {1,...,n} ε i, opìte B(x, ε) n i=1 U i. Sunep c, n i=1 U i T ρ. (GiatÐ de mporoôme na jewr soume aujaðretec tomèc? Diìti, an gia parˆdeigma tan ε n = 1 n, n N, tìte de ja up rqe katˆllhlo ε jetiko, afoô ε = inf n N ε n = 0.) (iii) 'Estw I mh kenì sônolo kai {U i } i I aujaðreth oikogèneia anoikt n uposunìlwn tou X. 'Estw x i I U i (diaforetikˆ, T ρ ). Tìte, upˆrqei toulˆqiston èna i 0 I tètoio, ste x U i0. AfoÔ U i0 anoiktì, upˆrqei ε > 0, ste B(x, ε) U i0 i I U i. 'Ara, i I U i T ρ. 'Askhsh 1.4. 'Estw sônolo X. (i) DeÐxte ìti h T = P(X) eðnai mða topologða tou X. H topologða aut kaleðtai diakrit topologða tou X. Epˆgetai apì kˆpoia metrik sto X? Me ˆlla lìgia, upˆrqei metrik ρ, ste ta anoiktˆ uposônola tou metrikoô q rou (X, ρ) na tautðzontai me ta stoiqeða thc topologðac T? (ii) DeÐxte ìti h T = {, X}, eðnai mða topologða tou X. H topologða aut kaleðtai tetrimmènh topologða tou X. Epˆgetai apì kˆpoia metrik sto X?

7 1.1. TOPOLOGŸIA 3 (iii) OrÐzoume mða sqèsh << >> merik c diˆtaxhc sthn oikogèneia ìlwn twn topologi n tou X, me T 1 T 2, an, kai mìno an, T 1 T 2. DeÐxte ìti upˆrqoun T min, T max wc proc th sqèsh thc merik c diˆtaxhc, ste gia kˆje T topologða tou X na isqôei T min T T max. (iv) An X = {a, b} kai T = {, X, {a}}, deðxte ìti h T eðnai topologða pou den epˆgetai apì metrik sto X. AkoloujoÔn kˆpoia paradeðgmata, pou apantoôn en mèrei kai sta erwt mata (i), (ii) kai (iv) thc ˆskhshc. ParadeÐgmata 1.5. 'Estw sônolo X. (i) H diakrit topologða T = P(X) epˆgetai apì th diakrit metrik ρ d. Apìdeixh. Sto m.q. (X, ρ d ), gia kˆje x X kai gia opoiod pote ε (0, 1], isqôei B(x, ε) = {x}. Sunep c, kˆje uposônolo A tou metrikoô q rou ja eðnai anoiktì, afoô A = x A {x}. Dhlad, T T ρ d. T ra, profan c kˆje anoiktì uposônolo tou metrikoô q rou eðnai uposônolo tou X, dhlad T ρd P(X) = T. Epomènwc, T = T ρd, kai ˆra o diakritìc topologikìc q roc eðnai, ìpwc alli c lème, metrikopoi simoc. (ii) H tetrimmènh topologða T = {, X} den epˆgetai apì kˆpoia metrik, ìtan to X perièqei perissìtera apì èna stoiqeðo, dhlad X 2 (ìpou me X sumbolðzoume ton plhjikì arijmì tou X). Apìdeixh. 'Eqoume upojèsei ìti to X eðnai mh kenì, sunep c X > 0. An X = 1, dhlad X = {x}, tìte, gia kˆje metrik ρ me thn opoða efodiˆzoume to X, isqôei T ρ = {, X}, diìti, afenìc to kai to X eðnai ta monadikˆ uposônola tou X, afetèrou eðnai kai ta dôo anoiktˆ se kˆje metrikì q ro (X, ρ). An X 2, tìte upojètoume proc apagwg se ˆtopo ìti upˆrqei metrik ρ sto X, ste na epˆgei thn T, dhlad T ρ = T. a' trìpoc 'Estw x 1 x 2 dôo stoiqeða tou X. Jètoume U = B(x 1, ε), ìpou ε = ρ(x 1,x 2 ) 2, kai parathroôme ìti x 1 U, x 2 U kai U T ρ. Upojèsame, ìmwc, ìti T = T ρ, ˆra eðte U = eðte U = X, to opoðo eðnai ˆtopo.

8 4 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI b' trìpoc Sto m.q. (X, ρ), kˆje monosônolo {x} eðnai kleistì, isodônama, to X {x} (pou eðnai diˆforo tou kenoô, diìti X 2) eðnai anoiktì gia kˆje x X. Epomènwc, X {x} T ρ gia kˆje x X, to opoðo eðnai ˆtopo, diìti upojèsame ìti T ρ = T. (iii) EÔkola diapist noume ìti h oikogèneia T = {X F : F X, F peperasmèno} { } eðnai mða topologða tou X. H T kaleðtai sumpeperasmènh topologða tou X. ParathroÔme ìti gia tuqaða metrik ρ, me thn opoða efodiˆzoume to X, eðnai T = {X F : F X, F peperasmèno} { } T ρ. Prˆgmati, an F peperasmèno uposônolo tou X, tìte eðnai kleistì wc proc th metrik ρ (afoô den èqei shmeða suss reushc), kai ˆra to X F ja eðnai anoiktì uposônolo tou metrikoô q rou X, dhlad X F T ρ. Epiplèon, parathroôme ìti, an to X eðnai peperasmèno, h sumpeperasmènh topologða tautðzetai me th diakrit. Autì prokôptei, diìti kˆje uposônolo tou X (opìte kai kˆje sumpl rwma uposunìlou tou X) ja eðnai peperasmèno. 'Estw, loipìn, F P(X). ParathroÔme ìti F = X (X F ) kai ìti X F eðnai peperasmèno. Epomènwc, to F ja an kei sth sumpeperasmènh topologða. Sunep c, ja èqoume P(X) T. 'Omwc, gia kˆje topologða T isqôei T P(X), ˆra diakrit kai sumpeperasmènh topologða tautðzontai. Efìson t ra tautðzetai me th diakrit topologða, sðgoura ja epˆgetai apì th diakrit metrik. 'Omwc, eðdame parapˆnw ìti gia kˆje metrik ρ èqoume T = {X F : F X, F peperasmèno} { } T ρ. Epomènwc, P(X) T ρ, kai profan c èpetai ìti P(X) = T ρ. Dhlad, ìtan to X eðnai peperasmèno, h sumpeperasmènh topologða epˆgetai apì kˆje metrik ρ sto X. (iv) An X = {a, b}, tìte h topologða T = {, {a}, X} den epˆgetai apì kˆpoia metrik. Prˆgmati, se kˆje m.q. (X, ρ) to X {a} = {b} ja eðnai anoiktì, dhlad {b} T ρ. Epomènwc, T T ρ.

9 1.2. BŸASEIS, UPOBŸASEIS Bˆseic, upobˆseic Orismìc 1.6. 'Estw (X, T ) t.q. (topologikìc q roc). MÐa upooikogèneia B thc T ja kaleðtai bˆsh gia thn T, an kˆje anoiktì uposônolo tou X ( alli c, kˆje stoiqeðo thc T ) grˆfetai wc ènwsh stoiqeðwn thc B. Dhlad, an gia kˆje U T, upˆrqei oikogèneia {B i } i I B, ste U = i I B i. Ta stoiqeða thc bˆshc B kaloôntai basikˆ (anoiktˆ) uposônola tou topologikoô q rou (X, T ). ParadeÐgmata 1.7. (i) H T eðnai mða bˆsh gia thn Ðdia thn T. (ii) Stouc metrikoôc q rouc gnwrðzoume dh ìti kˆje anoiktì sônolo grˆfetai wc ènwsh apì anoiktèc mpˆlec (ˆskhsh). An, loipìn, (X, ρ) m.q. kai T ρ h metrik topologða tou X, tìte h B = {B(x, ε) : x X, ε > 0} eðnai mða bˆsh gia thn T ρ. EÔkola diapist nei kaneðc ìti h B = {B(x, q) : x D, q Q + } eðnai exðsou mða bˆsh gia thn T ρ, gia kˆje D puknì uposônolo tou X (bl. Pragmatik Anˆlush). (iii) An X = R kai ρ h sun jhc metrik stouc pragmatikoôc, tìte mða bˆsh gia thn metrik topologða T ρ eðnai h B = {(a, b) : a < b kai a, b R}. Prˆgmati, oi mpˆlec B(x, ε) = (x ε, x + ε) tou metrikoô q rou (R, ρ ) brðskontai se amfimonos manth antistoiqða me ta diast mata (a, b) tou R. 'Ara, to zhtoômeno èpetai apì to parˆdeigma (ii). Lìgw thc puknìthtac twn rht n stouc pragmatikoôc arijmoôc, h upooikogèneia B = {(a, b) : a < b kai a, b Q} eðnai epðshc mða bˆsh gia thn T ρ. H epìmenh prìtash, dojèntoc enìc t.q. (X, T ), sunistˆ èna krit rio, pou apofaðnetai an mða upooikogèneia B thc T eðnai bˆsh gia thn T. Prìtash 1.8. 'Estw (X, T ) t.q. kai B T. H B eðnai bˆsh gia thn T, an, kai mìno an, gia kˆje U T kai kˆje x U, upˆrqei B B tètoio, ste x B U. Apìdeixh. ( ) 'Estw ìti h B eðnai bˆsh gia thn T. Epilègoume èna U T kai èna x U (an tan U =, tìte to zhtoômeno èpetai eôkola). Ex orismoô thc bˆshc, ja upˆrqei

10 6 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI sônolo I kai {U i } i I B, ètsi ste U = i I U i. Sunep c, ja upˆrqei i 0 I tètoio, ste x U i0 U. ( ) AntÐstrofa, èstw U T. Ex upojèsewc, gia kˆje x U upˆrqei B x B tètoio, ste x B x U. 'Ara, ja eðnai U = x U B x (giatð?), opìte sumperaðnoume ìti h B eðnai bˆsh gia thn T. Prìtash 1.9. 'Estw (X, T ) t.q., B mða bˆsh gia thn T kai U X. Ta epìmena eðnai isodônama: (i) To U eðnai anoiktì. (ii) Gia kˆje x U, upˆrqei B B tètoio, ste x B U. Apìdeixh. (i) (ii) To U, wc stoiqeðo thc T, ja grˆfetai U = i I U i gia kˆpoia oikogèneia stoiqeðwn {U i } i I thc bˆshc B. Opìte, an x U, upˆrqei èna i 0 I, ste x U i0 U. (ii) (i) Apì thn prohgoômenh prìtash, gia kˆje x U epilègoume B x B, ste x B x U. Epomènwc, ja eðnai U = x U B x, kai ˆra U T. Sto krit rio pou eðdame parapˆnw, jewr same dedomèno èna sônolo X, efodiasmèno me mða topologða T, kai exetˆsame th sunj kh pou prèpei na ikanopoieð mða tuqaða upooikogèneia B thc T, ste na eðnai bˆsh gia thn T. T ra ja jewr soume monˆqa èna sônolo X kai ja exetˆsoume poièc sunj kec prèpei na ikanopoieð mða tuqaða oikogèneia B uposunìlwn tou X, ste na eðnai bˆsh gia kˆpoia topologða tou X. PrwtoÔ proqwr soume sto je rhma - krit rio, ac parathr soume ìti, an h tuqaða B eðnai bˆsh gia kˆpoia topologða tou X, tìte h topologða aut ja eðnai ex orismoô h oikogèneia ìlwn twn en sewn stoiqeðwn thc B. Dhlad, mða bˆsh prosdiorðzei monos manta thn topologða sthn opoða eðnai bˆsh (en antijèsei me thn topologða, pou, en gènei, den èqei monadik bˆsh). Tèloc, epishmaðnoume ìti, apì ed kai sto ex c, gia na elègxoume thn kleistìthta peperasmènwn tom n, mporoôme na elègqoume monˆqa thn kleistìthta thc tom c dôo sunìlwn, kai apì epagwg ja èqoume to zhtoômeno.

11 1.2. BŸASEIS, UPOBŸASEIS 7 Je rhma 'Estw sônolo X kai B P(X). H B eðnai bˆsh gia kˆpoia topologða tou X, an, kai mìno an, èqei tic parakˆtw idiìthtec: (i) X = {B : B B}. (ii) Gia kˆje B 1, B 2 B kai x B 1 B 2, upˆrqei B 3 B, ste x B 3 B 1 B 2. Apìdeixh. ( ) 'Estw ìti h B eðnai bˆsh gia kˆpoia topologða T tou X. Tìte, (i) efìson X T, ja upˆrqei mða oikogèneia stoiqeðwn {B i } i I thc B tètoia, ste X = i I B i. 'Eqoume ìti EpÐshc, B i {B : B B}. i I {B : B B} T P(X). Sunep c to {B : B B} eðnai uposônolo tou X, kai ˆra X = {B : B B}. (ii) (ˆskhsh). ( ) 'Opwc èqoume parathr sei, an h B eðnai bˆsh gia kˆpoia topologða, tìte aut ja eðnai h oikogèneia ìlwn twn en sewn stoiqeðwn thc B. JewroÔme, loipìn, thn oikogèneia { } T = B i : I sônolo, B i B i I. i I Autì pou prèpei na deðxoume eðnai ìti, ìtan h B èqei tic idiìthtec (i) kai (ii), tìte h T eðnai mða topologða tou X. Opìte kai h B ja eðnai mða bˆsh gia thn T. Apì thn idiìthta (i), prokôptei ìti X T (X = i I B i, ìpou I = B kai B i = i). EpÐshc, gia I =, èqoume ìti T. H T eðnai kleist stic aujaðretec en seic. Prˆgmati, èstw {U j } j J T. Tìte, gia kˆje j J, ja eðnai U j = i I j B i. Sunep c, U j = ( j J j J ìpou I = j J I j. Epomènwc, j J U j T. i I j B i ) = B i, i I

12 8 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI H T eðnai kleist stic peperasmènec tomèc (jumhjeðte thn epis mansh pou kˆname ep' autoô). Prˆgmati, èstw U 1, U 2 T. Ja eðnai U 1 = i I 1 B i kai U 2 = i I 2 B i. Sunep c, U 1 U 2 = (B i1 B i2 ). (i 1,i 2 ) (I 1 I 2 ) H tom dôo stoiqeðwn thc B den an kei en gènei sth B. 'Omwc, apì thn idiìthta (ii) prokôptei ìti B i1 B i2 = x B i1 B i2 B x, ìpou B x B gia kˆje x B i1 B i2. Sunep c, B i1 B i2 T, kai ˆra U 1 U 2 T. 'Eqontac katˆ nou thn ènnoia thc grammik c j khc, thc kurt c j khc, akìmh kai thc kleistìthtac enìc uposunìlou metrikoô q rou, kaj c kai touc trìpouc perigraf c touc (eswterik kai exwterik perigraf ), tðjentai fusiologikˆ ta ex c erwt mata: 'Estw èna sônolo X kai mða oikogèneia F uposunìlwn tou X. Upˆrqei mða topologða tou X, pou na eðnai h mikrìterh topologða tou X pou perièqei thn F? An nai, p c mporoôme na thn kataskeuˆsoume? EÐnai monadik? Orismìc 'Estw X sônolo, T mða topologða tou X kai F P(X). Ja lème ìti h T parˆgetai apì thn F kai ja sumbolðzoume me T = T (F), an h T ikanopoieð tic parakˆtw idiìthtec: (i) F T. (ii) Gia kˆje topologða T tou X me F T, isqôei T T. Parathr seic (i) SÔmfwna me ton parapˆnw orismì, h T (F) eðnai h mikrìterh topologða tou X pou perièqei thn F. Wstìso, den èqoume apodeðxei akìmh thn Ôparxh thc. (ii) An upˆrqei h T (F), tìte eðnai monadik. Prˆgmati, an up rqe mða deôterh topologða T tou X pou na parˆgetai apì thn F, tìte, me diadoqik efarmog tou orismoô gia kˆje mða apì tic T kai T, prokôptei ìti T T kai T T. AkoloujeÐ h exwterik perigraf thc T (F), pou sugqrìnwc exasfalðzei kai thn Ôparx thc. Prohgoumènwc, af netai wc ˆskhsh na deðxete ìti, an {T i } i I P(P(X))

13 1.2. BŸASEIS, UPOBŸASEIS 9 eðnai mða oikogèneia topologi n enìc sunìlou X, tìte h i I T i eðnai mða topologða tou X (kat' analogða me touc dianusmatikoôc q rouc, ta kurtˆ kai ta kleistˆ sônola). Prìtash 'Estw X sônolo, F P(X), kai jewroôme thn topologða T = {C : C topologða tou X me F C}. H T eðnai h mikrìterh topologða tou X pou perièqei thn F, dhlad T = T (F). Apìdeixh. Arqikˆ, parathroôme ìti h topologða T eðnai kal c orismènh, diìti h oikogèneia {C : C topologða tou X me F C} perièqei to P(X), kai ˆra eðnai mh ken (an tan ken, tìte h T ja tan to sônolo ìlwn twn sunìlwn!). H T ex orismoô perièqei thn F. Tèloc, eðnai h mikrìterh pou perièqei thn F, diìti, an T topologða tou X me F T, tìte h T ja an kei sthn oikogèneia {C : C topologða tou X me F C}, kai sunep c T T. SuneqÐzoume me thn eswterik perigraf thc T (F), dhlad ton trìpo me ton opoðo, xekin ntac apì thn F, ja qtðsoume th mikrìterh topologða pou thn perièqei. Prìtash 'Estw X sônolo kai F P(X). OrÐzoume { n B(F) = i=1 H B(F) eðnai mða bˆsh gia thn T (F). } F i : n N {0}, {F i } n i=1 F {X}. Apìdeixh. Arqikˆ, ja deðxoume ìti h B(F) eðnai bˆsh gia kˆpoia topologða, èstw T. Autì ja gðnei me efarmog tou jewr matoc 1.10: ParathroÔme, kat' arqˆc, ìti B(F) P(X). EpÐshc, isqôei ìti X = {B : B B(F)}, diìti, afenìc B X gia kˆje B B(F), opìte {B : B B(F)} X, afetèrou X B(F), opìte X {B : B B(F)}. 'Estw B 1, B 2 B(F) kai x B 1 B 2. Ex orismoô h B(F) eðnai kleist stic peperasmènec tomèc, sunep c x B 1 B 2 B(F)

14 10 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI (sômfwna me to sumbolismì tou jewr matoc 1.10, ja eðnai B 3 = B 1 B 2 ). 'Ara, h B(F) eðnai mða bˆsh gia thn T. Profan c, h T kajorðzetai monos manta apì thn B(F): h T ja eðnai h oikogèneia ìlwn twn en sewn stoiqeðwn thc B(F). Mènei na deðxoume ìti T = T (F). Prˆgmati, èstw T topologða tou X pou perièqei thn F. H T eðnai kleist stic peperasmènec tomèc, kai ˆra B(F) T. 'Omwc, h T eðnai kleist kai stic aujaðretec en seic, sunep c T T. Epomènwc, T = T (F). Orismìc 'Estw (X, T ) t.q. kai F P(X). H F kaleðtai upobˆsh gia thn T, an T = T (F). Ta stoiqeða thc upobˆshc F kaloôntai upobasikˆ (anoiktˆ) uposônola tou topologikoô q rou (X, T ). Parathr seic 'Estw X sônolo kai F P(X). (i) Gia na kataskeuˆsoume (eswterik perigraf ) th mikrìterh topologða pou perièqei thn F, arkeð na mazèyoume se mða oikogèneia B(F) ìlec tic peperasmènec tomèc stoiqeðwn thc F, kai èpeita olec tic en seic stoiqeðwn thc oikogèneiac B(F). (ii) H F mporeð na eðnai pˆnta upobˆsh gia kˆpoia topologða tou X, kai sugkekrimèna gia thn T (F). Parˆdeigma H F = {(, x) : x R} {(y, + ) : y R} apoteleð upobˆsh gia th sun jh metrik topologða tou R. Den apoteleð, ìmwc, bˆsh, afoô to anoiktì diˆsthma (a, b) de mporeð na prokôyei wc ènwsh stoiqeðwn thc F, allˆ wc (a, b) = (, b) (a, + ). 1.3 Stoiqei deic ènnoiec thc topologðac Orismìc 'Estw (X, T ) t.q. kai F X. To F ja kaleðtai kleistì, an to X F eðnai anoiktì, dhlad an X F T. Prìtash 'Estw (X, T ) t.q. Tìte, h oikogèneia F twn kleist n uposunìlwn tou X ikanopoieð tic parakˆtw idiìthtec: (i) To kai to X eðnai kleistˆ. (ii) H F eðnai kleist stic peperasmènec en seic.

15 1.3. STOIQEIŸWDEIS ŸENNOIES THS TOPOLOGŸIAS 11 (iii) H F eðnai kleist stic aujaðretec tomèc. Apìdeixh. Af netai wc ˆskhsh (upìdeixh: orismìc tou kleistoô kai kanìnec De Morgan). Parathr seic 'Estw (X, T ) t.q. (i) 'Ena uposônolo tou X den eðnai kat' anˆgkh oôte anoiktì oôte kleistì. Me ˆlla lìgia, an F h oikogèneia twn kleist n uposunìlwn tou X, tìte isqôei T F P(X). (ii) Ap' thn ˆllh pleurˆ, eðnai dunatìn èna uposônolo tou X na eðnai anoiktì kai kleistì sugqrìnwc. DÔo tetrimmèna paradeðgmata aut c thc perðptwshc, pou isqôoun se kˆje topologikì q ro, eðnai to kai to X. 'Allo èna aplì parˆdeigma, eðnai autì tou diakritoô topologikoô q rou, ston opoðo kˆje uposônolo tou q rou eðnai anoiktì kai kleistì sugqrìnwc. ParadeÐgmata (i) An jewr soume to R me th sun jh metrik topologða T ρ, tìte ta dexiˆ kai ta aristerˆ hmianoiktˆ diast mata den eðnai oôte anoiktˆ oôte kleistˆ (apl eformog twn orism n). (ii) Ston (R, T ρ ) den upˆrqei A R, mh kenì, ste na eðnai anoiktì kai kleistì sugqrìnwc. Prˆgmati, upojètoume proc apagwg se ˆtopo ìti upˆrqei tètoio A. Tìte, to R A ja eðnai epðshc anoiktì. GnwrÐzoume ìti kˆje anoiktì uposônolo tou R grˆfetai wc arijm simh ènwsh anoikt n kai xènwn diasthmˆtwn. Sunep c, dedomènou ìti R = A (R A), mporoôme na grˆyoume to R wc arijm simh ènwsh anoikt n kai xènwn diasthmˆtwn, to opoðo eðnai ˆtopo, diìti ta ˆkra twn diasthmˆtwn aut n de ja an koun sto R. (iii) An jewr soume ton (R, T ρ ) kai Y = R {x 0 }, ìpou x 0 R, tìte h T Y = {U {x 0 } : U T ρ } eðnai mia topologða tou Y (ˆskhsh). ParathroÔme ìti (, x 0 ) T Y kai Y (, x 0 ) = (x 0, + ) T Y, dhlad to (, x 0 ) eðnai anoiktì kai kleistì sugqrìnwc. (iv) An F mða oikogèneia uposunìlwn enìc sunìlou X, tìte h F ikanopoieð tic idiìthtec thc prìtashc 1.19, an, kai mìno an, h T = {X F : F F} eðnai topologða tou X.

16 12 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI 'Askhsh H topologða T Y tou paradeðgmatoc (iii) entˆssetai sthn kathgorða thc sqetik c topologðac, h opoða sunistˆ ton pio aplì kai fusiologikì trìpo paragwg c enìc nèou topologikoô q rou apì ènan proôpˆrqonta: 'Estw (X, T ) t.q. kai Y X. DeÐxte ìti h oikogèneia T Y = {U Y : U T } eðnai mða topologða tou Y. H T Y kaleðtai sqetik topologða tou Y (wc proc thn T ). O Y efodiasmènoc me thn T Y kaleðtai upìqwroc tou X. Orismìc 'Estw (X, T ) t.q. kai A X. H kleistìthta tou A sumbolðzetai me A cl T A kai orðzetai wc to mikrìtero kleistì uposônolo tou X pou perièqei to A, dhlad ikanopoieð tic ex c idiìthtec: (i) to A eðnai kleistì, (ii) A A, (iii) an F X kleistì me A F, tìte A F. Prèpei na deðxoume ìti h kleistìthta enìc sunìlou eðnai èna kal c orismèno sônolo, dhlad ìti upˆrqei kai eðnai monadikì. Proc toôto, akoloujeð h epìmenh prìtash, pou exasfalðzei thn Ôparxh kai apoteleð th gn rimh exwterik perigraf thc ènnoiac pou orðsame. H monadikìthta, efìson exasfalðsoume thn Ôparxh, prokôptei me apl efarmog tou orismoô thc kleistìthtac (af netai wc ˆskhsh). Prìtash 'Estw (X, T ) t.q. kai A X. Tìte, A = {F : F kleistì uposônolo tou X me A F }. Apìdeixh. (ˆskhsh) Me thn epìmenh prìtash dðdoume th eswterik perigraf thc kleistìthtac enìc sunìlou.

17 1.3. STOIQEIŸWDEIS ŸENNOIES THS TOPOLOGŸIAS 13 Prìtash 'Estw (X, T ) t.q. kai A X. Tìte, x A, an, kai mìno an, gia kˆje U T me x U, isqôei U A. Dhlad, A = {x X : U A, gia kˆje U T me x U}. Apìdeixh. ( ) 'Estw x A. Ac upojèsoume proc apagwg se ˆtopo ìti upˆrqei U anoiktì me x U, ste na isqôei U A =. Tìte, A X U. EpÐshc, X U eðnai kleistì, sunep c, ex orismoô thc kleistìthtac, èqoume A X U. Epomènwc, afoô x A, prokôptei ìti x X U. Autì, ìmwc, eðnai ˆtopo, diìti ex upojèsewc x U. ( ) Pˆli upojètoume proc apagwg se ˆtopo ìti x A. Tìte, x X A, pou eðnai anoiktì. Ex upojèsewc, loipìn, èqoume ìti (X A) A, pou eðnai ˆtopo, afoô A A. Orismìc 'Estw (X, T ) t.q., A X kai x X. To x ja kaleðtai shmeðo suss reushc tou A, an isqôei U (A {x}) gia kˆje U T me x U. To sônolo twn shmeðwn suss reushc kaleðtai parˆgwgoc tou A parˆgwgo sônolo tou A kai sumbolðzetai me A. Orismìc 'Estw (X, T ) t.q., A X kai x A. To x ja kaleðtai apomonwmèno shmeðo tou A, an upˆrqei U T, ste U A = {x}. Parathr seic 'Estw (X, T ) t.q. kai A X. (i) ParathreÐste ìti, sômfwna me ton orismì, gia na eðnai èna x shmeðo suss reushc tou A, den apaiteðtai na an kei sto A. AntÐstrofa, an èna x an kei sto A, tìte de sunepˆgetai apì ton orismì ìti ja eðnai shmeðo suss reushc tou A. MporeÐte na epalhjeôsete sth sun jh metrik topologða tou R, gia A = { 1 n : n N} (poiì eðnai to A?). (ii) JewroÔme èna x pou an kei sto A. Tìte, to x eðnai shmeðo suss reushc tou A, an den eðnai apomonwmèno shmeðo tou A, kai antðstrofa, to x eðnai apomonwmèno shmeðo tou A, an den eðnai shmeðo suss reushc tou A. Shmei ste pwc h upìjesh en prokeimènw, dhlad ìti to x an kei sto A, epibˆlletai apì ton orismì, efìson jèloume na qrhsimopoi soume thn ènnoia tou apomonwmènou shmeðou.

18 14 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI (iii) IsqÔei ìti A A. Prˆgmati, an x A, tìte U (A {x}) U A gia kˆje U T me x U. Prìtash 'Estw (X, T ) t.q. kai A X. IsqÔoun ta parakˆtw: (i) A A, (ii) A = A, (iii) an A B, tìte A B, (iv) (A B) = A B, (v) to A eðnai kleistì, an, kai mìno an, A = A. Apìdeixh. (ˆskhsh) Prìtash 'Estw (X, T ) t.q. kai A X. IsqÔei ìti A = A A. Apìdeixh. 'Estw x A. EÐte x A eðte x A. An x A, tìte x A A. An x A, tìte, gia kˆje U T me x U, isqôei U (A {x}) = U A, kai ˆra x A A A. 'Estw x A A. An x A, tìte profan c x A. An x A, ìpwc eðdame sthn parapˆnw parat rhsh (iii), tìte x A. Pìrisma 'Estw (X, T ) t.q. kai A X. To A eðnai kleistì, an, kai mìno an, perièqei ta shmeða suss reus c tou. Apìdeixh. ( ) An A kleistì, apì thn prìtash 1.29 èqoume ìti A = A. Opìte, qrhsimopoi ntac thn prìtash 1.30, èqoume ìti A A A = A = A. ( ) An A A, tìte, qrhsimopoi ntac thn prìtash 1.30, èqoume ìti A = A A = A. Epomènwc, apì thn prìtash 1.29, to A eðnai kleistì. Pìrisma 'Ena sônolo pou den èqei shmeða suss reushc eðnai kleistì. Apìdeixh. 'Amesh efarmog tou porðsmatoc 1.31.

19 1.3. STOIQEIŸWDEIS ŸENNOIES THS TOPOLOGŸIAS 15 'Askhsh 'Estw (X, T ) t.q., A, B X kai {A i } i I P(X). (i) DeÐxte ìti A B A B. IsqÔei h isìthta? (Upìdeixh: Ston (R, T ρ ) jewr ste ta A = Q, B = R Q). (ii) DeÐxte ìti i I A i ( i I A i) gia kˆje sônolo I. IsqÔei h isìthta? H {A i } i I kaleðtai topikˆ peperasmènh, an, gia kˆje x X, upˆrqei U T, ste to sônolo {i I : U A i } na eðnai peperasmèno. DeÐxte ìti, an h {A i } i I eðnai topikˆ peperasmènh, tìte isqôei h isìthta i I A i = ( i I A i). (iii) DeÐxte ìti, an U T, tìte U A, an, kai mìno an, U A. Orismìc 'Estw (X, T ) t.q. kai A X. To eswterikì tou A sumbolðzetai me A int T A kai orðzetai wc to megalôtero anoiktì uposônolo tou X pou perièqetai sto A, dhlad ikanopoieð tic ex c idiìthtec: (i) to A eðnai anoiktì, (ii) A A, (iii) an U X anoiktì me U A, tìte U A. Prìtash 'Estw (X, T ) t.q. kai A X. Tìte, A = {U : U anoiktì uposônolo tou X me U A}. Apìdeixh. (ˆskhsh) Prìtash 'Estw (X, T ) t.q. kai A X. IsqÔoun ta parakˆtw: (i) (A) c = (A c ), (ii) (A ) c = (A c ), ìpou me A c sumbolðzoume to X A.

20 16 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI Apìdeixh. (i) (A) c = ( {F : F c T me A F }) c = {F c : F c T me A F } = {F c : F c T me F c A c } = {U : U T me U A c } = (A c ). (ii) (ˆskhsh). Prìtash 'Estw (X, T ) t.q. kai A X. IsqÔoun ta parakˆtw: (i) A A, (ii) (A ) = A, (iii) an A B, tìte A B, (iv) (A B) = A B, (v) to A eðnai anoiktì, an, kai mìno an, A = A. Apìdeixh. (ˆskhsh) 'Askhsh (i) DeÐxte ìti A B (A B). BreÐte èna antiparˆdeigma, gia na apodeðxete ìti den isqôei en gènei h isìthta. (ii) DeÐxte ìti, an A anoiktì, tìte A (A). BreÐte èna antiparˆdeigma, gia na apodeðxete ìti den isqôei en gènei h isìthta. 'Askhsh (i) 'Estw X ˆpeiro sônolo, T h sumpeperasmènh topologða tou X kai A X. BreÐte ta A kai A. (ii) 'Omoia gia X uperarijm simo sônolo, efodiasmèno me th sunarijm simh topologða T = {X A : A X arijm simo}.

21 1.3. STOIQEIŸWDEIS ŸENNOIES THS TOPOLOGŸIAS 17 Orismìc 'Estw (X, T ) t.q. kai U T. To U kaleðtai kanonikì anoiktì, an U = (U). Parat rhsh EÔkola mporeðtai na diapist sete ìti isqôei U (U). En gènei, den eðnai ìla ta anoiktˆ uposônola enìc topologikoô q rou kanonikˆ anoiktˆ. Gia parˆdeigma, ston topologikì q ro (R, T ρ ) gia U = (, 0) (0, + ) eðnai (U) = R U. 'Omwc, ìla ta diast mata thc morf c (a, b), (a, + ), (, b) eðnai kanonikˆ anoiktˆ. Orismìc 'Estw (X, T ) t.q. kai A X. To sônoro tou A sumbolðzetai me Bd(A) ( (A)) kai orðzetai wc Bd(A) = A (A c ). Parathr seic (i) Apì ton qarakthrismì thc kleistìthtac prokôptei ìti Bd(A) = {x X : U A kai U A c, gia kˆje U T me x U}. (ii) EpÐshc, isqôei ìti Bd(A) = Bd(A c ). Prìtash 'Estw (X, T ) t.q. kai A X. IsqÔei ìti Bd(A) = A A. Apìdeixh. Bd(A) = A (A c ) = A (A ) c = A A. Pìrisma 'Estw (X, T ) t.q. kai A X. IsqÔoun ta parakˆtw: (i) A = A Bd(A), me A Bd(A) =. (ii) X = A Bd(A) (A c ), me ta A, Bd(A), (A c ) na eðnai xèna anˆ dôo. Apìdeixh. (i) 'Amesh efarmog thc teleutaðac prìtashc. (ii) Efarmìzontac to (i) kai thn prìtash 1.36, èqoume ìti X = A (A) c = A Bd(A) (A) c = A Bd(A) (A c ). Parat rhsh 'Estw A èna uposônolo enìc topologikoô q rou X. To (A c ) kaleðtai exwterikì tou A kai sumbolðzetai me Ext(A).

22 18 KEFŸALAIO 1. TOPOLOGIKOŸI QŸWROI Epomènwc, to (ii) tou parapˆnw porðsmatoc mporeð na grafeð me diaforetikì sumbolismì wc X = int(a) Bd(A) Ext(A), me ta int(a), Bd(A), Ext(A) na eðnai xèna anˆ dôo. Me ˆlla lìgia, kˆje uposônolo A diamerðzei to q ro X se trða xèna mèrh: sto eswterikì tou A, sto sônoro tou A kai sto exwterikì tou A.

23 Kefˆlaio 2 SuneqeÐc sunart seic Orismìc 2.1. 'Estw (X, T ), (Y, S) topologikoð q roi kai f : X Y sunˆrthsh ( apeikìnish). H f ja kaleðtai suneq c, an f 1 (U) T gia kˆje U S. UpenjÔmish 2.2. 'Estw X, Y sônola, f : X Y sunˆrthsh kai U Y. H antðstrofh eikìna tou U mèsw thc f sumbolðzetai me f 1 (U) kai orðzetai wc f 1 (U) = {x X : f(x) U}. H antðstrofh eikìna mèsw thc f (pou eðnai mða sunolosunˆrthsh me pedðo orismoô to P(Y )) den prèpei na sugqèetai me thn antðstrofh sunˆrthsh thc f. H taôtish aut mporeð na gðnei, mìno ìtan h f eðnai 1-1 kai epð. JumhjeÐte, epðshc, ìti h f 1 sumperifèretai kalˆ stic sunolojewrhtikèc prˆxeic. ParadeÐgmata 2.3. (i) H tautotik apeikìnish Id : (X, T 1 ) (X, T 2 ) eðnai suneq c, an, kai mìno an, T 2 T 1. (ii) MÐa sunˆrthsh f : (X, T ) (Y, S) kaleðtai anoikt, an f(u) S gia kˆje U T. MÐa sunˆrthsh mporeð na eðnai anoikt, allˆ ìqi suneq c. 'Ena tètoio parˆdeigma eðnai h tautotik apeikìnish Id : (X, T ) (X, P(X)), ìpou T P(X). Prˆgmati eðnai anoikt, diìti f(a) P(X) gia kˆje A X, opìte kai gia kˆje U T. Den eðnai ìmwc suneq c, diìti f 1 (U) = U T gia kˆje U (P(X) T ). (iii) Pˆntote mporoôme na broôme toulˆqiston mða suneq sunˆrthsh metaxô dôo topologik n q rwn. Elègxte ìti oi stajerèc sunart seic ikanopoioôn to zhtoômeno. 'Askhsh 2.4. 'Estw (X, T ), (Y, S) t.q. DeÐxte ìti an h f : X Y eðnai suneq c, 19

24 20 KEFŸALAIO 2. SUNEQEŸIS SUNARTŸHSEIS tìte kai h f : X f(x) eðnai suneq c, ìtan o f(x) eðnai efodiasmènoc me th sqetik topologða. Orismìc 2.5. 'Estw (X, T ), (Y, S) t.q. (i) MÐa sunˆrthsh f : X Y kaleðtai omoiomorfismìc, an eðnai 1-1, epð, suneq c, kai h f 1 : Y X eðnai suneq c. Se aut thn perðptwsh lème ìti oi X, Y eðnai omoiomorfikoð ìti o X eðnai omoiomorfikìc me ton Y, kai sumbolðzoume me X Y. (ii) MÐa sunˆrthsh f : X Y kaleðtai omoiomorfik emfôteush, an o X eðnai omoiomorfikìc me ton f(x) pou eðnai efodiasmènoc me th sqetik topologða. Se aut thn perðptwsh lème ìti o X emfuteôetai omoiomorfikˆ ston Y, kai sumbolðzoume me X f(x) me X Y. Parathr seic 2.6. (i) An h f : X Y eðnai omoiomorfik emfôteush epð tou Y, tìte profan c eðnai omoiomorfismìc. (ii) An mða sunˆrthsh f metaxô dôo topologik n q rwn eðnai 1-1, epð kai suneq c, tìte h f 1 den eðnai kat' anˆgkh suneq c. Elègxte ìti h tautotik apeikìnish Id : (X, P(X)) (X, T ), ìpou T P(X), eðnai mða tètoia perðptwsh. (iii) Parathr ste ìti apì topologik skopiˆ dôo omoiomorfikoð q roi tautðzontai. Prˆgmati, ta stoiqeða touc, allˆ kai ta anoiktˆ uposônola touc brðskontai se amfimonos manth antistoiqða (ˆskhsh). Sthn topologða, ousiastikˆ, meletˆme ekeðnec tic idiìthtec pou mènoun analloðwtec mèsw omoiomorfism n; oi idiìthtec autèc kaloôntai topologikèc. 'Askhsh 2.7. (i) 'Estw (X, T ), (Y, S) t.q. kai f : X Y sunˆrthsh 1-1 kai epð. DeÐxte ìti h f eðnai omoiomorfismìc, an, kai mìno an, eðnai suneq c kai anoikt. (ii) 'Estw o topologikìc q roc R, efodiasmènoc me th sun jh metrik topologða, kai o ( 1, 1) me th sqetik topologða. DeÐxte ìti h sunˆrthsh f : R ( 1, 1) me f(x) = x 1+ x eðnai omoiomorfismìc.

25 21 Diapist ste ìti h idiìthta thc plhrìthtac den eðnai topologik (upìdeixh: o metrikìc q roc ( 1, 1) den eðnai pl rhc). (iii) DeÐxte ìti h sqèsh omoiomorfismoô sthn klˆsh twn topologik n q rwn eðnai sqèsh isodunamðac. Prìtash 2.8. 'Estw (X, T ), (Y, S) t.q. kai f : X Y sunˆrthsh. Ta epìmena eðnai isodônama: (i) H f eðnai suneq c. (ii) To f 1 (F ) eðnai kleistì uposônolo tou Y, gia kˆje F kleistì uposônolo tou X. (iii) Gia kˆje x X kai gia kˆje U S me f(x) U, upˆrqei W T me x W, ste f(w ) U. (iv) f(a) f(a) gia kˆje A X. (v) f 1 (B) f 1 (B) gia kˆje B Y. Apìdeixh. ProtoÔ proqwr soume sthn apìdeixh, zhtoôme apì ton anagn sth na epalhjeôsei genikˆ ìti, an A X kai B Y, tìte f(f 1 (B) B kai A f 1 (f(a)), qwrðc na alhjeôoun kat' anˆgkhn oi isìthtec kai stic dôo sqèseic. (Se poiˆ perðptwsh isqôei h isìthta se kˆje mða sqèsh?) (i) (ii) ArkeÐ na parathr soume ìti (X f 1 (F )) = f 1 (X F ). (i) (iii) 'Estw x X kai U S me f(x) U. Tìte, x f 1 (U), kai lìgw sunèqeiac f 1 (U) T. An jèsoume W = f 1 (U), èqoume to zhtoômeno, kajìti f(w ) = f(f 1 (U) U. (iii) (i) 'Estw U S. An f 1 (U) =, tìte f 1 (U) T. An f 1 (U), epilègoume x f 1 (U), opìte f(x) U. Tìte, upˆrqei W x T me x W x, ste f(w x ) U, kai ˆra W x f 1 (f(w x )) f 1 (U). Epomènwc, f 1 (U) = x f 1 (U) W x, to opoðo profan c eðnai anoiktì wc ènwsh anoikt n. (ii) (iv) 'Estw A X. ParathreÐste ìti f(a) f(a), an, kai mìno an, A f 1 (f(a)). Epomènwc, arkeð na deðxoume ton deôtero egkleismì. Profan c, f(a) f(a), opìte

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