The Use of Filters in Topology

Transcription

1 The Use of Filters i Topology By ABDELLATF DASSER B.S. Uiversity of Cetral Florida, 2002 A thesis submitted i partial fulfillmet of the requiremets for the degree of Master of Siee i the Departmet of Mathematis i the College of Arts ad Siees at the Uiversity of Cetral Florida Orlado, Florida Fall Term 2004

2 ABSTRACT Sequees are suffiiet to desribe topologial properties i metri spaes or, more geerally, topologial spaes havig a outable base for the topology. However, filters or ets are eeded i more abstrat spaes. Nets are more atural extesio of sequees but are geerally less friedly to work with sie quite ofte two ets have distit direted sets for domais. Operatios ivolvig filters are set theoreti ad geerally ertai to filters o the same set. The oept of a filter was itrodued by H. Carta i 1937 ad a exellet treatmet of the subjet a be foud i N. Bourbaki (1940). ii

3 ACKNOWLEDGEMENTS would like to express my gratitude to my supervisor, Dr. Rihardso, for may isightful oversatios durig the developmet of the ideas i this thesis, his uderstadig, edless patiee, ad eouragemet. would also like to thik my defese ommittee members, Dr. Mohapatra ad Dr. Ha for askig me exellet questios. Fially, but ot least, would like to thik Ahmed Ameur for the friedship ad omi relief whe thigs started to get diffiult. iii

4 TABLE OF CONTENTS CHAPTER 1 NTRODUCTON AND EXAMPLES... 1 CHAPTER 2 FLTERS... 4 CHAPTER 3 ULTRAFLTERS... 8 CHAPTER 4 CONVERGENCE AND FLTERS CHAPTER 5 COMPACTNESS AND FLTERS CHAPTER 6 NTAL STRUCTURES CONCLUSON LST OF REFERENCES iv

5 CHAPTER 1 NTRODUCTON AND EXAMPLES The study of filters is a very atural way to desribe overgee i geeral topologial spae. Filters were itrodued i 1937 by Carta (1937 a,b). Bourbaki (1940) employed filters i order to prove several results i their text. the same year Tukey (1940) studied sets, filters, ad various modifiatios of the two oepts. A omplete reliae o filters for the developmet of topology a be foud i Kowalsky (1961). There are traes of the oept of filters as early as 1914 i Root s artile. More reetly, filters play a fudametal role i the developmet of fuzzy spaes whih have appliatios i omputer siee ad egieerig. Filters are also a importat tool used by researhers desribig o-topologial overgee otios i futioal aalysis. (e.g see Beattie ad Butzma,2002). Moreover, Preuss (2002) has applied filters throughout his book o ategorial topology. The purpose of this paper is to provide thorough disussio of filters ad their appliatios. Filters are used i geeral topology to haraterize suh importat oepts as otiuity, iitial ad fial strutures, ompatess, et. The followig examples are give to show that sequees are ot suffiiet to haraterize poits of losure, otiuity, ad ompatess. Example 1.1 let X to be a uoutable set ad fix x X 0. Defie { A X : ( x A) or ( x Aad A is outable) } (a) φ, X τ τ. The τ is a topology for X. = 0 0 1

6 (b) A, B τ implies that A B τ () A τ, J implies that A τ. The latter holds sie if x0 A for some 0 0, the A = A A 0 whih is outable. Thus, A τ. We laim that x τ x if ad oly if x = x evetually ( ie = x N ). x (a) Suppose that x. Sie{ x } τ, x τ x if ad oly if x 0 (b) Suppose that x = x0 ad x 0 for ifiitely may. Defie F= x = x evetually. x { x x } : x 0. The F τ ad x F evetually fails to hold. Hee x x0 ifiitely ofte. Therefore, x 0 does ot overge to x. Coversely, if does ot overge to x, the there exist x 0 O τ, x 0 O suh that x O ifiitely ofte. That is, x x0 ifiitely ofte. Hee, x x 0 if ad oly if x = x τ 0 evetually. A: { x } but it does ot exist a sequee { x } i { } x0 0 { } x 0 suh that x τ x 0. f x, the x x for all 1 ad by above results, x does ot overge to x 0 0 x. Hee, there is o sequee otaied i { x } 0 that overges to. However, 0 { x } sie O x 0 for eah O τ, x 0 O. Ad therefore, sequees do ot x0 0 { } φ haraterize poits of losure. B: Let σ be the disrete topology for X, ie σ is the set of all subsets of X. Oe a see x 0 that x x 0 if ad oly if x = x evetually. Hee, σ ad τ have the same σ 0 overget sequees. Let the d: (, τ ) ( X,σ ) X deote the idetity futio. The 2

7 futio d is sequetially otiuous sie τ ad σ have the same overget sequees. However, sie τ σ, the above futio is ot otiuous. Hee, sequees do ot haraterize otiuity. Example1.2 Let ( X, d ) be a metri spae that is ot ompat. Ad let ( X *,τ * ) be the Stoe-Ceh ompatifiatio of X ( d ) (, d). Sie X, is ot sequetially ompat, there exist { x } otaied i X whih has o overget subsequee i ( X, d). t is kow that o sequee otaied i X overges to a poit i X * X. Hee, ( X *,τ * ) is ot sequetially ompat. Therefore, sequees do ot haraterize ompatess. 3

8 CHAPTER 2 FLTERS Defiitio 2.1 Cosider a arbitrary set X. A set τ of subsets of X satisfyig the oditios: (a) φ τ ad X τ (b) U V τ wheever u τ ad V τ () The uio of the members of a arbitrary subset of τ belogs to τ is alled a topology o X. A topologial spae is a pair ( X,τ ) where τ is a topology o X. The members of τ are alled ope sets. Defiitio 2.2 Cosider a set X φ. A filter F o X is a set of subsets of X satisfyig the oditios: (a) F φ ad φ F (b) f () f B F the A, A B A F ad A B X the B F F A subset β F is alled a base for the filter F if every member of F otais some member of β. The defiitio of a filter base for some filter is as follows: Defiitio 2.3: β is alled a base for a filter o X if ad oly if β is a set of subsets of X satisfyig the oditios: (a) β φ, φ β 4

9 (b) B 1, B2 β B3 β suh that B3 B1 B 2. Example 2.4 Let X φ be a arbitrary set. Fix x0 X ad the x& 0 = { A : A X ad x0 A} is a filter o X. x 0 x& Note that { 0}. Example 2.5 Fix a set φ A0 X, the 0 = A & { B X B } : A 0 is a filter o X. partiular, if A0 = X, X & ={ X} is the smallest possible filter o X. Example 2.6 f X is ay oempty set ad { x} is a sequee i X. Defie B = { x k : k 1}. The, F = { A X : A B 1} is a filter o X ad is alled the elemetary filter determied by { x }. Example 2.7 f X is a ifiite set the F={ F X : F is fiite} a filter o X ad is alled the ofiite or Fréhet filter. Example 2.8 f X is a topologial spae ad x X, the the family U(x) of all eighborhoods of x is a filter ad is alled the eighborhood filter of x. Example 2.9 The family of all tails of the sequee { x } o X is a base for the orrespodig elemetary filter; a tail is the set of the form = N {{}} Example 2.10 The family x is a base for the filter x& o X. B { x : N}. Let F (X) deote the set of all filters o a set X ad F, G F (X). We all a filter G fier tha the filter F if F G, we also all F oarser that G. Note that F = {X} is the oarsest member i F (X). t is easy to verify that ( F (X ), ) is a poset. 5

10 Also, F( X, ) is ot liearly ordered sie x& y& or y& x&. Before disussig filter ad overgee, oe wats to prove ad defie various thigs about filters. DeMorga s law states that if X φ ad{ A : } is a olletio of subsets of X. The: (a) ( A ) = i A ( ) (b) A = i A Propositio 2.11 Assume that X φ ad F F(X ), (idex set) the F F(X ) is the fiest filter o X whih is oarser tha eah F,. Proof (a): Note that φ F,, implies that φ F ad also X belogs to eah F,. F Thus X ad it follows that F φ. (b): Let A, B F ; the A ad B belog to eah F. Therefore, A B belogs to eah F,, ad thus A B F. (): Let A F ; the A eah F,. Let B A thus B belogs to eah F sie B is a over set of A. t follows that B F. Sie (a), (b) ad () are satisfied, F is a filter. Clearly F F, for eah. 6

11 Next, let G F for eah ad let us prove that G F. Let A G; thus A belogs to eah F,, ad the A F. t follows that G F. geeral, the uio of two filters may or may ot be a filter. For example, if F ad G may otai disjoit members. Propositio 2.12 Let F,, be filters o X. The F ={ A X : A = F } F for some F Proof Let B F ; the B belogs to eah F ad thus B = F by hoosig eah F = B. Coversely, Let B { A X : A = F for some F F }, thus B = F for some F F ad thus B belogs to eah F. Hee B F Letφ X Y. f F F(X ), the F is a filter base o Y. That is { A Y : F A F F } is a filter o Y geerated by F. The geerated filter is deoted by [F ]. Coversely, if G is a filter o Y ad G X φ for eah G G, the F = { G X : G G} F(X). This filter is alled the idued filter o X, or the trae of G o X. Example 2.13 Let X = [0,1], Y = R ad let G be the filter o Y whose base is {( εε, ): ε> 0}. The the trae of G o X is the filter o X havig a base {[ 0, ]: ε > 0 } ε. 7

12 CHAPTER 3 ULTRAFLTERS (, ) Defiitio 3.1 A ultrafilter is a maximal filter i the poset F( X),where the orderig F G meas that F G. That is a filter U o X is a ultrafilter provided U G implies that U =G. Propositio 3.2 ( Zor s lemma ) f X is partially ordered set i whih every liearly ordered subset ( ay two elemets are omparable ) has a upper boud, the X has a maximal elemet. That is, there exists x X suh that there is o y x with x y. Propositio 3.3 Let X be a set ad F a filter o X. The there exists a ultrafilter U o X that is fier tha F.. Proof: Cosider the family P={ G F ( X ) : Gis a filter that fier tha F }. The family P { G } is partially ordered by. Suppose that C = liearly ordered subset of P for eah G P. : Deote H= { G : } ={ A : A G } (a) F P by ostrutio, thus H φ (b) φ H sie φ G, for eahg P is a hai i P. That is C is ( ) Let A, B H; the A Gβ, B G β for some, β. Now, either Gβ G or Gβ G holds. The A, B Gβ ad thus A B G β sie Gβ is a filter. The, A B H. (d) f A H the A G for some G C. f B A the B G ad thus B H. 8

13 Therefore, H is a filter ad hee a upper boud for C i P The partially ordered set P satisfies the assumptios of Zor s lemma; hee there is a maximal elemet U P. Therefore, U is a ultrafilter otaiig F. Propositio 3.4 Let F be a filter o a set X; the, the followig are equivalet: (a) F is a ultrafilter. (b) For ay two subsets A ad B of X we have: f A U B ) For every subset A of X either A F or A F. Proof (a) (b): Assume A U B F ad A F ad B F. F the A F or B F. Defie G = { C X : A C F }; the G F(X).Further, F G, B G ad thus F G. But F is a ultrafilter. Thus, there is a otraditio. Therefore, A F or B F. (b) (): Clearly A A = X F ; therefore A F or A F usig(b). () (a): Let G be a filter that is fier tha F ad let A G be arbitrary. The A F beause A has oempty itersetio with every elemet of F. t follows that A F. Thus, G = F. Propositio 3.5 Let F F(X). The: F = { U : U F U, is aultrafilter o X } Proof Clearly F { U : U F U, is aultrafilter o X }. Assume that there exists a { } A U : U F U, is aultrafilter o X suh that A F. The F A φ for eah F F ad thus β = { F A : F F } is a base for a filter G. Note that F G. 9

14 Let UG be a ultrafilter otaiig G. Sie, F UG, A U G. However, A G U G ad thus A A U, a otraditio. Therefore, G F = U : U F U, is aultrafilter o X. { } Propositio 3.6 Let U be a ultrafilter o the set X. f AA..., 1 2 A are subsets of X suh that A belogs to U, the at least oe of the sets belogs to U. =1 i i A i A i Proof f o belogs to U, the A belogs to U for i i = 1,2,..., by Propositio 2.3, ad hee A = ( A ) belogs to U, whih is impossible sie A is give to belog to U.. i i =1 i i Propositio 3.7 let U be a ultrafilter o X ad A U A φ for all U U. The A U. X suh that Proof Assume that A U ad defie β = { A U U } φ β ad β φ sie U A ( U1 A) ( U 2 A)= ( U U 2 ) A β φ for all U U. Also, U :. Note that 1 sie ( U 1 U 2 ) U. Thus β is a base filter for some filter G.Note that A G beause A U A ad A U. Therefore, U G, a otraditio ad hee A U. Propositio 3.8 Let f : X Y be a futio ad U is a ultrafilter o X. The f (U ) is a ultrafilter o Y. Proof Assume that ( ) A f U ; the for eah U U, A f ( U ) φ. 10

15 1 Hee, f ( A ) U φ 1 Therefore, f ( f ( A ) f ( U ) is a ultrafilter o Y. 1 for eah U U ad thus by Propositio 2.6, f ( A ) U. ad thus f ( U ) A 1 sie f ( f ( A )) A. Hee, f (U) 11

16 CHAPTER 4 CONVERGENCE AND FLTERS Defiitio 4.1 Let ( X,τ ) be a topologial spae ad let U (x) deote the eighborhood filter at x. A filter F o X overges to x if U (x) F. Propositio 4.2 Let A be a subset of a topologial spae X. The, for x X, x Ā if ad oly if there exists a filter o X whih otais A ad overges to x. Proof Assume that x Ā. The ay eighborhood of x has a oempty itersetio with A. Now all the sets A U, where U is a eighborhood of x, form a filter base, ad the orrespodig filter overges to x. Coversely, assume that F is a filter otaiig A ad overgig to x. Choose ay eighborhood U of x. The U F, ad thus U A φ sie A F. This proves that x Ā. Defiitio 4.3 A topologial spae ( X, τ ) is Hausdorff or T 2 provided if x y, the there exist sets O x, O y τ suh that x Ox, y Oy ad O x O y = φ. Propositio 4.4 ( X, τ ) is T if ad oly if eah filter overges to at most oe poit, i.e. 2 F τ x, y implies x=y. Proof Suppose that ( X, τ ) is Hausdorff ad suppose F τ x, y where x y. The there exist O x, O y τ suh that x Ox, y Oy ad O x O y = φ.however, F τ x, y implies that O F, x Oy F, ad 12

17 Ox O y =φ F whih is a otraditio. Therefore, eah filter overges to at most oe poit. Coversely, suppose that x y ad assume that O O φ for eah x y O x, O y τ, x Ox, y Oy We laim that β = O O : O τ, O τ, x O, y O } is a base for some filter { x y x y x y F.Observe that ( Ox O y ) ( G x G y ) = ( Ox Gx ) ( Oy G y ) β sie ( Ox Gx ) is a ope set otaiig x ad O G ) is a ope set otaiig y. Thus β is a base ( y y for some filter F sie O O O ) implies that eah Ox F, x ( x y F overges to x. Likewise, Oy ( Ox Oy ) implies that Oy F, ad thus F overges to y, a otraditio. Therefore, there does t exist a O ad O suh thato where x ad y O. Hee ( X, τ ) is Hausdorff. O x y x y x O The followig result shows that otiuity of maps betwee topologial spaes a be haraterized i terms of overgee of filters. Reall that a mappig g: X Y betwee two topologial spaes is otiuous at x provided that for eah eighborhood V of g ( x ), there exist a eighborhood U of x suh that g(u) V. Propositio 4.5 Let X, Y be topologial spaes with x X ad y φ g: X Y. The g is otiuous at x if ad oly if wheever F is a filter suh that F x, g(f ) g(x). Proof Suppose g is otiuous at x ad F x. Let V be a eighborhood of g(x). By otiuity there is a eighborhood U of x suh that g(u) V. Sie U F, 13

18 g(u) g(f ). Ad thus V g(f ). Hee g (F ) g(x). Coversely, suppose that wheever F x, g(f) g (x). The g( U ( x) ) g( x) by hypothesis. The, for eah eighborhood V of g (x), V g ( U ( x) g (U) V ad thus g is otiuous at x. ). The there exists a U U(x) suh that 14

19 CHAPTER 5 COMPACTNESS AND FLTERS Reall that a topologial spae ( X,τ ) is ompat provided eah ope overig of X has a fiite suboverig. t is show below that ompatess a be haraterized i terms of overgee of ultrafilters. Propositio 5.1 Let ( X, τ ) be a topologial spae. The the followig statemets are equivalet: (a) ( X, τ ) is ompat (b) Eah ultrafilter o X overges () { A: A } F φ for eah filter o X Proof (a) (b): Suppose ( X, τ ) is ompat ad that there exist a ultrafilter F that does t overge. The for eah x X, there exists F V V ad thus C = { V : x X} x x is a ope over of X. Hee V i= xi 1. =X. Sie F is a ultrafilter, implies that Vx F for eah x X ad therefore X = φ F, whih is a otraditio. = = i 1 Vx i Thus, eah ultrafilter o X overges. (b) (): Give ay filter F o X; let G be a ultrafilter otaiig F. The G overges to x i ( X, τ ), for some x X. Give ay eighborhood V of x ad A F; the A G ad V G. Hee, A V G ad thus A V φ. Therefore, x A. 15

20 () (a): Suppose ( X,τ ) is ot ompat. Let C ={ O J} : be a ope over of X with o fiite subover. The, = i 1 O i X, for eah. Let F be the filter o X whose base O, i = i 1 : 1 O whih is a otraditio. C.However, { A : A F } O = ( O ) = X φ, = J Propositio 5.2 Let f : ( X, τ ) ( Y, σ ) be a otiuous futio ad oto. f ( X, τ ) is ompat the ( Y, σ ) is ompat. 1 Proof Let U be a ultrafilter o Y, F = ( U ) f ad by Propositio2.2 there exist a ultrafilter o X. G F. The: G τ x, for some x X sie ( X, τ ) is ompat. The otiuity assumptio of f implies that f (G ) f(x) aordig to Propositio 4.6. Sie G F the f (G) f ( F ) =f (f 1 ( U)) U. Also f is oto ad thus 1 f (f (U))= U. Hee, f(g ) U ad sie U is a ultrafilter f ( G ) = U. Cosequetly, f (G ) f (x) implies that U f(x). Therefore, by Propositio 4.1, ( Y, σ ) is ompat. 16

21 CHAPTER 6 NTAL STRUCTURES f Propositio 6.1 Cosider the soure X ( Y ), J The:, where J, is a idex lass. τ (a) There exists a oarsest (smallest) topology τ o X for whih eah f :( X, τ ) ( Y, τ ) is otiuous, J. (b) Eah g: Y, σ ) ( X, τ ) is otiuous if ad oly if f ο g ( Y, σ ) ( Y, τ ) is ( otiuous, for eah J. () τ is the uique topology for X whih obeys (b) ( ) (d) Give F F X, τ x if ad oly if f τ F ( F ) f ( x ) i : for eah J. Proof A subbase for τ is S ={ f ( O ) : O } 1 τ. (a) t easily follows that τ is the oarsest suh topology suh that eah f is otiuous. (b) The ompositio of two otiuous futios i otiuous. Coversely, assume that 1 f ο g is otiuous, for eah J. Let f ( O ) S the, 1 )) = ( fο g) ( O ) σ ad thus, ( ) σ g ( f ( O g f ( O ) Sie otiuity of g: Y, σ ) ( X, τ ) ( follows that g: Y, σ ) ( X, τ ) is otiuous. ( is determied by the subbase S for τ, it 17

22 () Let τ be aother topology for X obeyig (b). Sie id: X, τ ) ( X, τ ) is X ( X otiuous, f ( X, τ X ) ( Y, τ ) is otiuous, for eah J. Hee by (a), τ τ X. : Moreover, osider id: X, τ ) ( X, τ ). ( X Sie f f ο id ( X, τ ) ( Y, τ ) = is otiuous for eah J, the hypothesis : implies that id: ( X, τ ) ( X, τ X ) is otiuous. Hee, τ X τ ad thus τ X = τ. (d) Sie eah f :( X, τ ) ( Y, τ ) f τ ( f ( x). Coversely, assume that is otiuous, F τ x implies that. τ F) f ( ) f ( x) F for eah J. Now, 1 f ( O ) S ad if x f 1 ( O ), f ( x) O f ( F) ad thus f ( F ) O for some F F 1. Hee f ( O ) F ad thus 1 f ( ) F. Sie f 1 ( ) is a base O = i 1 i O i member for τ, it follows that base members otaiig x belog to F. τ Hee F x. partiular, whe X ΠX is a produt set ad = P f = are projetio maps, τ is alled the produt topology for X. Aordig to Propositio 5.1 τ is the oarsest topology for X suh that eah projetio map is otiuous. 18

23 CONCLUSON A disussio of filters ad their appliatios to the theory of geeral topology has bee preseted. These ideas have bee primarily developed by Europea mathematiias, begiig with the work of H. Carta ad reorded i the various texts writte uder the pseudoame N. Bourbaki. The Moore- Smith et overgee has bee more widely taught i Ameria uiversities. Both are used to haraterize topologial oepts i more abstrat spaes. 19

24 LST OF REFERENCES Beattie, R. ad Butzma, H.-P., Covergee struture ad appliatios to Futioal aalysis, klwwer aod. Pub., Dordreht, Bourbaki, N.(1940), Topologie geerale. Chapitre 1 et 2. Atualites Si. d., 858 Carta, H.(1937a), Theorie des filters, ompt. Red., 205, Carta, H.(1937b), Filtres et ultrafiltres, Compt. Red., 205, Kowalsky, H. J.(1961), Topologishe Raume, Basel-Stuttgart (Topologial Spaes, Aademi Press, New York, 1964 ). Preuess, Gerhard, Foudatios of Topology, kluwer aod. Pub., Dordreht, Root, R.(1914), terated limits i geeral aalysis, Am. J. Math., 36,