A GENERALIZED PROCEDURE FOR DEFINING QUOTIENT SPACES b y HAROLD G. LAWRENCE A THESIS submitted to OREGON STATE UNIVERSITY in prtil fulfillment of the requirements for the degree of MASTER OF ARTS June 1963
APPROVED: L I ' V L Professor of Mti'l'i'(mt'ic s In Chrge of Mjor vn1rmn or 1V11:nemucs ~runent Chirmn of SchUGrdu&' Committ;;"""' / Den of Grdute School Dte thesis is presented August 3, 1962 Typed by Crol Bker
ACKNOWLEDGEMENT The uthor is indebted to Dr. B. H. Arnold for the suggestion of the subje ct of this pper nd for his helpful comments C.ur ing its writing.
TABLE OF CONTENTS CHAPTER Pge I. INTRODUCTION 1 II. ORIGIN OF PROBLEM 2 III. PRODUCT AND QUOTIENT SPACES 4 IV. GENERALIZED PROCEDURE 14 V. EXAMPLES 22 BIBLIOGRAPHY 26
A GENERALIZED PROCEDURE FOR DEFINING QUOTIENT SPACES Chpter I INTRODUCTION In the study of ny concept one my ttempt to generlize the concept so tht for specil cse of the gener.:...lizlon the originl concept is obtined. This is done with the hope tht the generliztion will led to some interesting results. It is the object here to generlize the procedure for defining the quotient topology nd quotient spce. In chpter two the problem will be outlined nd in chpter three those definitions nd theorems will be given which will be needed in chpter four in giving the generlized definitions. Finlly, some exmples will be given.
2 Chpter II ORIGIN OF PROBLEM Let X nd Y be sets nd f: X-7 Y be trnsformtion. From topology for X topology for Y my be defined reltive to f nd from topology for Y topology for X my be defined reltive to. The detils re presented in Chpter III. The cse of defining topology on the crtesin product ~S of fmily of topologicl spces {s} indexed by set A, reltive to the projection mppings p : II s ---7 sb b A nse s in the definition of product spce nd is treted in Chpter III of J. L. Kelley's, "Generl Topology" (2, p. 84-100}. In forming the product spce there is given fmily of trnsformtions p with common domin, II S. Ech p is mde A continuous by giving the domin II S the pproprite topology. A The quotient topology hs been defined on set Y reltive to topologicl spce X nd single trnsformtion f: X----7Y
3 so tht Y hs the lrgest topology such tht f is continuous. The following questions, which re not treted by Kelley, rise. Given fmily {s) of topologicl spces nd fmily of trnsformtions onto common rnge Y for ech in n index set A, cn Y be topologized with the lrgest topology such tht ll the f continuous? 1 so, is this procedure then generliztion of the procedure for defining the quotient topology? Is it possible to obtin from the fmily {s} of topologicl spces single topologicl spce X together with single trnsformtion f, obtined from the fmily (f} of trnsformtions, which determines exctly tht sme topology for Y tht mkes ll the f continuous? re It is the purpose of this pper to generlize the procedure for defining the quotient topology by showing how set Y my be topologized with the lrgest topology such tht, given the collections {f} nd {s}. ech of the trnsformtions f is continuous nd to nswer the bove questions.
4 Chpter III PRODUCT AND QUOTIENT SPACES In this chpter the concepts of product nd quotient spce will be defined nd those definitions nd theorems which will be needed in the following chpters will be given. Definition 1. A topology is fmily T of subsets of set S. The members of T re clled open sets nd stisfy the xioms: 0 1 The u nion of ny number of open sets is n open set. n open set. 0 The intersectior: of ny finite number of open sets is 2 0 Both Snd the empty set, (/J, re open. 3 The set S is clled the spce of the topology T nd the pir (S, T) is clled topologicl spce. Where the topology is understood, we my spek of the topologicl s pce S. Definition 2. A collection of subsets { B ) of given spce S is bsis for topology in S provided tht
5 1. UB =Snd, 2. If p is point of B nb, then there is n element B b ~ c of {B) which contins p nd is contined in BnBb. The topology for which { B } is bsis is obtined by letting s et be open in S if nd C' n] y if.it i ueion. of ele ments of Definition 3. A collection { U } of subsets of spce S is clled subbsis for topology in S provided tht the fmily of ll finite intersections of members of { U } is bsis for topology in S. Definition 4. A trnsformtion f of topologicl spce X into topologicl spce Y is continuous if nd only if for U open in Y, C 1 (U) is open in X. Definition 5. A home 0morphism is continuous one-to-one trnsformtion f of topologicl spce X onto topologicl spce Y such tht f - l is lso continuous. Given fmily ( S } of topologicl spces indexed by
set A, the crtesin product IIAS is the set of ll functions f on A 6 such tht f () is in S for ech in A. The set ITAS is the spce of the topologicl spce known s the product spce. In order to give the topology of the product spce, the projection mpping is defined by rr s ~s P b A b p (f) ::.: f(b). b - 1 The fmily of ll sets of the form pb (U) for b in A nd U open in S is then tken s subbsis for topology in II S According b A to Definition 3, the fmily B - { 0 l 0 is finite intersection of sets of the form must stisfy the conditions of Definition 2 to be bsis for topology in IIAS is stisfied, let U = S, then To see tht the first condition tht UB = II S A but for every f in 1'I S, ccording to d e finition, p (f) =f () where A f() is ins. Hence, since
7-1 - 1 p (U) =p (S ) = II S A is in B, UB = II S Let 0 nd 0 be in B, 0 n0 is lso finite A 1 2 1 2 intersection; hence, 0 no is in Bnd it follows tht the s econdcon 1 2 clition i s stisfied nd tht B is bsis for topology for TI S. A Definition 6. Given fmily of spces { S ) indexed by set A, the product spce is the crtesin product II S A with the topology for which subbsis is the collection of ll sets of the form pb -l (U), u open in sb. topology. The topology of the product spce is clled the product Definition 7. The quotient topology for the rnge Y of trnsformtion f defined on topologicl spce X is the fmily Q = { U I U is subset of Y such tht C 1 (U) is open in X}. Tht the fmily Q is topology follows from the fct tht the inverse of union of n rbitrry fmily F of subsets of Q is the union of the inverses of the members of F nd since the union
of the inverses of the members off is open in X, the union of the members of F is open in Y. Thus the fmily Q stisfies xiom 0 1 Since the inverse of n intersection of members of fmily F of subsets of Q is the intersection of the inverses of the members of F, iff hs finite number of members, the intersection of the 8 inverses of the members off is open in X nd hence the intersection of the members of F is open in Y. Thus the fmily Q stisfies xiom 0 Since f-l (Y) = X which is open in X, Y is 2 member of Q. The empty set is open in Y since the set of ll x in X such tht f (x} is in the empty set is empty nd hence open in X. Thus 0 3 is stisfied nd Q is topology. Let the rnge Y of trnsformtion f, defined on topologicl spce X, hve the quotient topology Q nd consider the fmily { f- 1 (y )) for y in Y of disjoint subsets of X. Now the union of the members of { r l(y)} is X. Let ~ = {d -1 } I d = f (y}' for some y in Y, then~ is decomposition of X into disjoint sets which cover X. Theorem l. If the projection mpping p :. X--~~ is defined by
-1 P (x) = f (f (x)) nd Ll is given the quotient topology reltive to P nd X, 9 then the topologicl spce.:lis homeomorphic to the topologicl spce (Y, Q). A homeomorphism is given by -1 g (y) =f (y). Proof: l. Let d be ny member of Ll. Then - l d ::.: f (y) for some y in Y. For tht y, - l g (y) = f (y) = d 0 Therefore g is mpping onto Ll. 2. Let y nd y be in Y with y =l-y 1 2 1 2 Since f is single vlued, - l -1 g (y l) = f (y l ) {; f (y 2) = g (y 2) Tht is g(y )-/; g(y ). 1 2 (Y, Q) onto Ll. Therefore g is one-to--one mpping of 3. Let U be open in Ll. -1 g (U) = ( y j g (y) is in U} = ( y I f-l (y)is contined in p-l (U)} -1 l but P (U) is open in X nd hence g- (U) is open in Y. Therefore g is continuous.
4. Let U be open Y, then g (U) :::: f-l (U) is open in X nd 10-1 hence is open in~ lso. Therefore g is continuous nd g is homeomorphism. Definition 8. The topologicl spce A is. clled the quotient spce reltive to P nd X. The term decomposition spce is sometimes used 1n plce of quotient spce. The concept of disjoint union of topologicl spces to be defined now will be needed. Definition 9. Given fmily {s} of topologicl spces, indexed by some set A, disjoint union of the fmily {S} is the spce UAT where for ech, T is homeomorphic to S nd for :/: b, T nt is empty. The spce U T b A is topologized by tking s bsis the union of the topologies for the topologicl spces T Tht the collection B = { u I U is open 1n some T ) is bsis follows from the fct tht T is open in the topologicl
spce T for e ch nd hence the union of ll the members of B is equl to U T 'A nd the first condition of Definition 2 is fulfilled. If U nd V re ny two members of B, then unv is empty unless U nd V re subsets of the sme spce T In this cse unv is open 11 by xiom 0 2 nd it follows tht condition 2 is fulfilled. Hence, B is bsis for topology in U T A Definition 10. The topology for UAT dete rmined by the bsis B is clled the disjoint union topology. If the given fmily {S} is lredy disjoint fmily, then for ech, T my be just S. If, on the other hnd, the fmily {s} is not disjoint, the fmily ( T )my be obtined by the following proce dure. For ech, {} is topologicl spce in which the only open sets re the empty set nd {). The cross product of S nd {). denoted by sx {}' yields fmily of topologicl spces SX {} = T for which, if does not equl b, T ntb is empty. It is seen tht for ech, T is homeomorphic to S Thus if set U is open ins, the set UX {} is open in T nd conversely.
Tht the union of n rbitrry fmily { S ) of topologicl spces, mking set open if it is the union of sets ech open in some S, is not necessrily topologicl spce is seen in the following 12 exmple. Let S 1 be the open intervl (0, 2} with the indiscrete topology (S nd the empty set re the only open sets) nd s be the 1 2 open intervl (1, 3) lso with the indiscrete topology. Then (0, 2)n(l, 3):: (1, 2) which is not open in (0, 2)U(l, 3) nd hence xiom 0 2 fils to be stisfied. Another theorem which will be needed is the following. Theorem 2. The intersection of ny number of topologies for spce S is topology for S. Proof: For ech in A, let 0 be topology for the spce S. Consider the intersection n 0. A l. The union of n rbitrry collection C of members of n 0 ~ is member of the intersection since the members of the collection C re present in ech 0 members of C is member of ech 0. nd hence the union of the 2. The intersection of finite collection C of members of
nao is member of nao since the members of C re present 13 in ech 0 nd hence the finite intersection of the members of C is member of ech 0 3. S nd the empty set re in ech 0 ; hence, S nd the empty set re in n 0 A definition. We re now in position to proceed with the generlized
14 Chpter IV GENERALIZED PROCEDURE Let collection { S ) of topologicl spces indexed by set A be g~ven 3.nd for e.c L in A, t rnsformticm S ---7Y from S onto the common rnge Y. If G denotes the quotient topology for Y reltive to f nd S, then by Theorem 2, G = n G A is topology for Y. Thus (Y, G) is topologic 2. l spce. Since G is the lrgest topology (hs the most open sets) for Y s uc h tht f is continuous function from S onto Y, it follows tht G is the lrgest topology for Y such tht ech of the functions f 1s continuous function. In this sense one might think of G s being the quotient topology reltive to the fmilies {s) nd{). Certinly if {S) nd {f} re singletons, then G is the quotient topology s previously defined. Thus we mke the following de inition. Definition 11. Let f S ----1 Y be trnsformtion nd G the quotient
topology on Y reltive to f nd the topologicl spce S for ech in the index set A, then the ~G is clled the quotient topology reltive to the fmilies {f) nd {s) 15 Theorem 3. Let J be the colle ction of ll sets of the form f -l (O) where 0 is in G, then J is topology for the spce S. Proof: If { ub) is n rbitrry collection of sets of the form f -l (Ob), Ob in G, then is open in (Y' G), U{ob) is open lll (Y' G). Hence - 1 is set of the form f (0), 0 in G, nd is therefore If { U,.., U } is finite collection of sets where 1 n - 1 U. = f (0.), 0. n element of G, then the 1 1 1 n n u = n. n f - 1 (0.) i =1 i 1= 1 1 = f - 1 ( n. n (0 ) ). 1= 1 i Since 0 is open in (Y, G) for i = 1,..., n, the nn (0.) is open in i i= l 1
16 (Y,G) nd hence then n U is in J. S i= 1 i nd the empty set re m J. S is in J since Y is in G nd f - l (Y) = S. The empty set is - 1 in J since it is in G nd f of the empty set is empty. It is thus seen tht the collection of sets J stisfies the three xioms 0, 1 Definition 12. The topology J is clled the G-dependent topology for S. It might be noted lso tht not only is J topology for S, but it is subtopology of the given topology for S, for if 0 is open - l in Y, f (0) is open in S with the originl topology since f is c ontinuous. Let us form the disjoint union over A of the fmily {(s, J )) of topologicl spces nd let T then the collection of ll sets ux{) such tht U is in J is bsis for topology T, Mle disjoint union topology, in U T A While the procedure for obtining the quotient topology G reltive to the fmilies {f) nd {s) is new, tht sme topology might be obtined from the trnsformtion
17 defined by f (x, ) = f (x). The motivtion for using the disjoint union is gin seen through the mnner in which the function f is defined on U T. A For not equl to b, S nd Sb might hve some point s in common, where f (s) = y nd fb (s) = y nd y does not equl y. The use 1 2 1 2 of the disjoint union gurntees tht f is single vlued, since (s, ) nd (s, b) re distinct points in U T, for t. A lest the second elements of the ordered pirs re different. Theorem 4. If Y is given the quotient topology Q reltive to f nd ( UAT, T), the topologicl spces (Y,Q) nd (Y,G) re ctully the sme spce. Proof: - 1 Let U be open in (Y, G). f (U) is open in (S, J ) for ech in A. Thus the U (f -l (U)X {}) is open in U T. Since A A U (f - 1 (U)X {)), A U is open in (Y, Q) nd therefore, G is subset of Q. -1 Secondly, let U be open in (Y, Q), then f (U) is open m U T A nd
18 for some subset B of A nd V ope n in (S, J ); tht is, C 1 (U) is union of bsis sets in the disjoi n t union. Since f is mpping onto Y, f ( U (V X {) )) = U. B But UBf (V ). - l V, being in (S, J ), is set of the form f (O) where 0 is - 1 member of G. Hence f (f (0)) = 0 is in G; t ht is, f (V ) is in G for ech in B nd consequently U f (V ) = U is in G. Q is therefore B subset of G nd the topologic l sp c es (Y, G) nd (Y, Q) re identicl. Consider the fmily ~ = { dj d = U A ( - l(y)x{}) for yin Y). Now~ is fmily of disjoint s ubsets of UAT which cover UAT. These sub sets re tken s the points in the spce ~ nd~ is given the quotient topology r e ltive to the projection p' U T-'-4 ~ A
19 defined by In cse T - 1 tht d = U f (y} for y in Y. A 1 P'(x,} = U (f- (f (x}}xf$). A = S, we could tke!:::j. 1 s the set of ll d such Definition 13. The topologicl spce A' is clled the quotient spce reltive to the fmily of trnsformtions f S ------7 Y. Theorem 5. If!:;,. is the quotient spce reltive to the trnsformtion f: U T ~ Y, then A is the sme topologicl spce s A'. A Proof: Since A is the quotient spce reltive to the trnsformtion f: UAT ----1 Y, ~hs the quotient topology reltive to the projection defined by P: U T---?A A P (x, ) = f- 1 (f (x, ) ) A set U is open in A if nd only if P-l (U) is open m UA T ' P(x,}:::; C 1 (f(x,)}isin l~. A set U is open in A 1 if nd only if P'- 1 (U} is open m U T. A
p -l(u)= { (x,) I P'(x,) = 20 U (f -l (f (x) )X {} ) is in u}. A But f - 1 (f (x, ) ) = C 1 (f {x) ) = 1 Hence A nd A re the sme topologicl spce. In nlogy with Theorem 1, we hve the following theorem. Theorem 6. The quotient spce A' reltive to the fmily of trnsformtion f: S---? Y from S onto Y is homeomorphic to the spce Y with the quotient topology G reltive to the fmilies {f} nd Proof: The quotient spce A is homeomorphic to (Y, Q) by Theorem 1, but A is A 1 by Theorem 5 nd (Y _, Q) is (Y, G) by Theorem 4. Suppose {f} nd {s}re singletons, (i.e., A = {}), reltive to the single trns- then the quotient topology G formtion f : S ~ Y is the quotient topology G reltive to the fmilies { f } nd { S } The G-dependent topology J is just the originl topology for S. We my let T = S in forming the disjoint union. In this cse then,
------------------------------------------------------ J is the disjoint union topology. f =f nd U T = S so tht A the quotient spce relti.ve to the fmilies {f} nd {s} is the quotient spce reltive to the single trnsformtion f : S --jy. Thus it is seen tht for the singletons {f} nd {S} the generlized procedure gives the sme results s the procedure previously defined for single trnsformtion. The generlized procedure for defining the quotient topology nd tbe quotient spce reltive to given fmily of trnsformtions f : S ~y is summrized in the following steps. 1. Determine the quotie nt topology G for Y reltive to f nd S for ech in A. 2. Determine the quotient topology G reltive to the 21 G = n G A 3. Determine the G-dependent topology J for ech S. 4. Determine the disjoint union topologicl spce 5. Determine the quotient spce!::i' reltive to the fmily of trnsformtions f: S---) Y.
22 Ch p ter V EXAMPLES Exmple l. A = { x I xis re l n u mber} S == E 1, the rel numbers with the topology (clled the usul topology of the rel numbers) b sis for which is the set of ll open intervls. Y = A ech in A. f : S ~y is given by f (x) ::: x for ech in A. l. (Y, G ) = E 1 for ech in A. 2. G = flg is the u s p-l topology. 3. The G-dependent topology J is the usul topology for 4. Let T = (S, J)X{}. UA T = {(r, s) I r nd s re rel numbers} Thus bsis for the disjoint union topology T is the set of ll U such tht for some rel numbers r, s, nd t, with r < s, U = { (x, t ) I r < x < s ).
5. A'= {d I d = l1_(f- 1 (x)x(}) for x in Y} 23 1 UA(f- (x}x{}) = {(x,} I is rel number} P': (UAT' T)---}A'is given by I 1 {} P (x,} = UA(f (f(x}x ) = { (x, ) I is rel number } A' with the quotient topology reltive to P' nd ( U T, T) A is the quotient spce reltive to the fmilies {f} nd { sj. rel number. f: ( UA T ' T}----+Y is given by f(x, ) =f (x) = x for ech 1 f- (x) = { (x, ) I is rel n urnbe r. } P: ( U T, T)~ A is given by A P (x, ) = C 1 ( (x, )) = C 1 (x). A with the quotient topology reltive to P nd ( U AT' T) is seen to be the sme spce s A 1 A bsis for the quotient topology in A' is the collection of ll subsets D of A of the form spce (Y,G) = El. D = { d I d = f-l (x) where c::: x< b} The topologicl spce A is thus homeomorphic to the
It is observed tht the topology T for the plne hs s subtopology the usul topology of the plne. 24 Exmple 2. A=={l,2} S == {, b, c} 1 The topology for S 1 is the fmily s 2 =={d,e,h} { ~,s, {}}. 1. The topology for s 2 is the fmily Y = { r, t} {Q),S 2, {d), { d,e), {h}, {d,~}. f : S ~y is given by 1 1 f ()= r, f (b) == t, f (c} = t. 1 1 1 f2: s2~ y is given by 2 (d} = r, 2 (e} = r, (h) = t. 2 1. G ={Q),Y,{r}} 1 G ={Q),Y,{r),(t}} 2 2. G=G ng ={p,y,{r}} 1 2 3. The G~dependent topology J l is the set {Q), s, {}} 1 The G-dependent topology J 2 is the set { Q), s, { d, e}}. 2
25 4. disjoint. T ={Cb.,s 1, s 2, {}, { d, e}, {, d, e}, {, b, c, d, e}, {, d, e, h}, {, b, c, d, e, h}} is the disjoint union topology. = {{,d,e}.{b,c,h}} P': ( U T, T) ~ ~ is given by A P' () = P 1 (d) = P' (e) = {, d, e} nd P'(b) = P ' (c) = P'(h) = {b,c,h). The quotient topology in~ 1 is the fmily {~ ~'.{,d,e}}. f: ( U T, T) ~ Y is given by A f() = f(d) = f(e) = r nd f(b) = f(c) = f(h) = t. Thus ~= {{,d,e), {b,c,h}} =~' Thus it is seen tht the topologicl spce ~is homeomorphic to the spce (Y, G).
26 BIBLIOGRAPHY 1. Hocking, John G., nd Gil S. Young. Topology. Reding, Msschusetts, Addison-Wesley, 1961. 374 p. 2. Kelley, John L. Generl Topology. Princeton, New Jersey, D. Vn Nostrnd, 1955. 298 p.