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Isomorphisms between semigroups of maps Warren, Eric 1972

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ISOMORPHISMS BETWEEN SEMIGROUPS OF MAPS by ERIC WARREN B.Sc, Dalhousie University, Halifax, Nova Scotia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Dep ar tmen t of MATHEMATICS We accept this thesis as conforming to the required standard The University of British Columbia April 1972 In present ing th i s thes is in pa r t i a l fu1 f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying o f th i s thes i s fo r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i c a t i on o f th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un ivers i t y of B r i t i s h Columbia Vancouver 8, Canada A B S T R A C T Let X and I be t o p o l o g i c a l spaces and C and .£ semigroups under composition of maps from X to X and Y to Y r e s p e c t i v e l y . Let H be an isomorphism from C to .5; i t i s shown that i f both C and J} contain the constant maps then there e x i s t s a b i j e c t i o n h from X to Y such that H ( f ) = hof-h"" 1, V f e C . Ve i n v e s t i g a t e t h i s s i t u a t i o n and T i n d s u f f i c i e n t -conditions f o r t h i s h to •be .a horneomorphism. In t h i s regard we study the f a m i l i a r semigroups of continuous, closed, and connected maps. An a u x i l i a r y problem i s the case when C = and H i s an automorphism of J), Ve then ask when i s every automorphism i s in n e r . The question i s answered f o r c e r t a i n p a r t i c u l a r semigroups? e.g., the semigroup of d i f f e r e n t i a b l e maps on the r e a l s has the property that a l l automorphisms are inner.' A CKNO VLEDGE1- !EN TS The A u t h o r i s i n d e b t e d t o D r . Sam B. N a d l e r , J r . f o r f i r s t i n t r o d u c i n g h i m t o t h e t o p i c s o f t h i s t h e s i s , a n d f o r h i s c o n t i n u e d e n c o u r a g e m e n t t h r o u g h o u t . He w o u l d a l s o l i k e t o t h a n k D r . A.H. C a y f o r d f o r h i s c r i t i c a l r e a d i n g o f t h e d r a f t c o p y , a n d D r . J.V. V h i t t a k e r f o r h i s e n l i g h t e n i n g comments a s w e l l a s h i s r e a d i n g o f the- d r a f t c o p y . He i s g r a t e f u l f o r t h e f i n a n c i a l a s s i s t a n c e p r o v i d e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a . F i n a l l y he w i s h e s t o t h a n k M i s s B a r b a r a K i l b r a y f o r h e r c a r e and c o n c e r n i n t y p i n g t h i s t h e s i s . T a b l e of C o n t e n t s I n t r o d u c t i o n D e f i n i t i o n s P r e l i m i n a r y Lemmas T — a d m i s s i b i l i t y A u t o m o r p h i s m s o f T(X) C o n c l u s i o n s R e f e r e n c e s ISOMORPHISMS BETWEEN SEMIGROUPS OF MAPS 1. I n t r o d u c t i o n . I f X i s a t o p o l o g i c a l s p a c e , then by X we denote - X .the set' o f a l l maps from X t o X . Then (X- ,°) i s a semigroup under composition.; we w r i t e f ( g ( x ) ) as (f°g)(x) . In the f o l l o w i n g we d i s c u s s . c e r t a i n subsemigrpups of X^ . Suppose X i s a g i v e n t o p o l o g i c a l space and T ( f ) i s a statement about f e X . Then by T(X) we denote X {f e X : T ( f ) } , and we c o n s i d e r o n l y t h o se T's s u f f i c i e n t t o make T(X) a subsemigroup o f X^ . , A c l a s s o f t o p o l o g i c a l s paces i s s a i d t o be a T - a d m i s s i b l e 1 c l a s s i f f o r e v e r y p a i r X , Y i n the c l a s s f o r which t h e r e i s an isom o r p h i s m H : T(X) -> T(Y) , then 3 h : X Y , such t h a t h i s a homeomorphism and H ( f ) = h°foh 1 , \/ f e T(X) . We n o t e t h a t f o r our purposes by isomorphism we w i l l always mean a one-to-one, onto homomorphism. We then have a c r i t e r i o n f o r two spaces b e i n g homeomorphic; i . e . , i f X and Y a r e b o t h T - a d m i s s i b l e and t h e r e e x i s t s some isomorphism from T(X) o n t o T(Y) , the n X and Y are homeomorphic. In f a c t , t h e r e a r e at l e a s t as many homeomorphisms from X t o Y as isomorphisms from T(X) t o T(Y) . We w i l l i n v e s t i g a t e T - a d m i s s i b i l i t y f o r c e r t a i n f a m i l i a r semigroups; e.g., the semigroup o f c o n t i n u o u s maps, w r i t t e n as S(X) ; and we w i l l g i v e examples o f q u i t e l a r g e T - a d m i s s i b l e c l a s s e s . A n o t h e r problem t o be c o n s i d e r e d i s t h e case o f automorphisms; V i z . , when i s e v e r y automorphism o f T(X) an i n n e r automorphism. We n o t e t h a t an automorphism, H , on a semigroup, T(X) , i s s a i d t o be i n n e r i f f ~3 h e X^ such t h a t h i s a b i j e c t i o n ( i . e . , i s i n v e r t i b l e ) w i t h b o t h h and h - 1 i n T(X) and such t h a t H ( f ) = hofoh" 1, V f e T(X) . 1 T h i s d e f i n i t i o n o f T - a d m i s s i b i l i t y i s ad a p t e d from t h a t of S - a d m i s s i -b i l i t y d e f i n e d by M a g i l l i n [3]. 2. Definitions. Let 'X be a topological space, then X Definition 2.1. If x e X , then by [x] we denote that map in X which is constantly x ; i.e., V x e X , [x](y) = x , V y e X . Also let K(X) = {[x] : x e X} . Definition 2.2. r (X) = {f e X X : f _ 1 ({x}) is closed, Vx e X> . Note: we are not being consistent here - T(X) need not be a semigroup. Definition 2.3. S(X) = {f e X : f is continuous} . Definition 2.4. C(X) = {f e X : f is closed; i.e., i f A is a closed subset of X , then so is f(A)} . X Definition 2.5. U(X) = (f e X : f is connected; i.e., i f A i s connectec then so i s f(A)} . 3. Preliminary. Lemmas. We now prove the following simple lemmas. First we note that an element, z , of a semigroup, V , is called a left-zero of V i f f zd = z , Vd e V . Lemma 3.1.2 If K(X)CT(X) , then K(X) i s precisely the set of left-zeroes of T(X) Proof. This follows from the following c i r c l e of arrows: f i s a'^left-zero of T(X) => f » [ x ] = f ,Vx e X => (fo ! x J)(y) = f(y) ,Vx,y e X => f(x) = f(y) ,\/x,y e X = > f e K(X) => fog = f ,\/g e T(X) => f is a l e f t zero of T(X) 2 Lemmas 3.1, 3.2, and 3.3 are a l l straightforward generalizations of parts of the discussion and proof of theorem 2.1 of [4]. 3. Lemma 3.2. If K ( X ) C T ( X ) and K(Y) <T T(Y) and H : T(X) -*• T(Y) i s an isomorphism, then H maps K(X) b i j e c t i v e l y onto K(Y) Proof. This follows from the following l i s t of equivalent statements: f e K(X) <=> fog = f , Vg e T(X) , by Lemma 3.1 <=> H(fog) = H(f) o H(g) -= H(f) ,Vg e T(X) , <=> H(f) e K(Y) , by Lemma 3.1 Lemma 3.3. Under the hypotheses of Lemma 3.2, there e x i s t s a unique b i j e c t i o n h : X -> Y such that H(f) = h ° f ° h _ 1 , \ / f E T(X) . Proof. Define h by h(x) = y i f f H([x]) = [y] . Then, by Lemmas 3.1 and 3.2, h i s a well-defined b i j e c t i o n . Now we simply note that Vy e Y and Vf e T(X) we have: (hofoh~1)(y) = h ( f ( h - 1 ( y ) ) ) , by d e f i n i t i o n of "o" , = [ h ( f ( h _ 1 ( y ) ) ) ] C y ) = H ( [ f ( h - 1 ( y ) ) ] ) ( y ) , by d e f i n i t i o n of h , = H( fo[h _ 1(y)])(y) = (H(f)°H([h 1 ( y ) ] ) ) ( y ) , H i s a homomorphism, = (H(f)o[y])(y) , by d e f i n i t i o n of h , = H(f)(y) . Hence H(f) = h°f°h 1 ,Vf £ T(X) . Uniqueness follows e a s i l y since i f k i s another map from X + Y such that H(f) = k 0 f ° k _ 1 ,\f£ c T(X) , then V x £ X we would have: H([x]) = ho[x]°h 1 = [h(x)] = k° [x] <>k 1 = [k(x)] => h = k . Now because of Lemma 3.3 we make the.following conventions: a) From now on a l l T(X) w i l l be such that K(X) ^ T ( X ) , for every topological space X b) If H : T(X) -> T(Y) is an isomorphism, then by h we denote the map from X to Y such that H(f) = hofoh"1 ,Vf e T(X) . That i s , we t a c i t l y use Lemma 3.3 and write the bijection that corresponds to an isomorphism as the lower case .latin letter of the upper case l a t i n letter which represents the isomorphism. The usefulness of Lemma 3.3, after we abide by the above conven-tions, then becomes clear. To determine whether a certain class i s T-admissible we merely have to show that a l l such h's are homeomorphisms. Similarly, an automorphism, H , is inner i f f h and h 1 are both in the domain of ;H . 4. T-Admissibility. We f i r s t concern ourselves with the space S(X) and the question of S-admissibility. We define below and S-space and show that S-spaces are S-admissible. We then give examples to show how large the class of S-spaces i s . Let X be a topological space, then. 3 Definition 4.1. If x e X and x e G , G an open set in X , then we say that G i s an S-neighbourhood of x i f either a) G = {x} or b) there exists a continuous f : G -»- X such that f(x) ={= x but f (y) = y , V y e G ~ G . Note: By G we mean the closure of G , and by G we mean the B closure of G relative to B O X . So G = G 3 The discussion of S-spaces and hence results 4.1 through to 4.6.2 are due to Magill in [5], We say that. X i s an S-space i f X i s Hausdorff and every.point i n X has a basis of S-neighbourhoods. This basis i s called an S-basis. Theorem 4.3. S-spaces are S-admissible.. Proof. Let X and Y be two S-spaces and suppose H : S(X) S(Y) is an isomorphism, then we need only show that h i s a homeomorphism. To do this we need only show that h and h 1 are closed, and to do this we only require that they be closed on basic closed sets. Because of this we f i r s t prove the following lemma. Lemma 4.4. If X is an S-space and <J>(f) = {x e X : f(x) = x} , Vf £ S(X) , then {cp(f) '• f e S(X)} is a basis for the closed sets of X . Proof. It i s well known that the set of fixed points of a continuous function i s closed, hence <|)(f) i s closed, \/f e S(X) . Now let F be a closed subset pf X . Then Vx e X ~ F , there i s an S-neighbour-hood, G , of x such that G X ~ F ; and there i s a continuous function q^: G -> X such that <l x( x) f x but ^ ( y ) = Y >Vy £ G ~ G (\(y) » y e G Define ^ : X - X by. g^y) =j y , y £ x . £ ' Then clearly \ g i s continuous. Also F C<j>(g ) ,V x e X ~ F . x r x ^» Hence F.C f\ ^(g^) . . But x \ <J,(gx),Vx E X~F => (X - F) f| (X(X1F xcX~F = 0 => (f,(g ) C F . So, in fact, F = ( ) <j>(g ) . Hence these . x eX~F " xeX~F X sets form a basis for the closed sets. 6. Now back to the theorem. Let f e S<X) , then <j>(H(f)) is closed. And y e <J>(H(f)) <=> ( h o f o h - 1 ) ( y ) = y <=> f ( h _ 1 ( y ) ) = h'^Cy) <=> h _ 1(y) e <j,(f) <=> y e h(<j>(f)) • Hence <j>(H(f)) = h(<j>(f)> ,Yf e S(X) . So h(<j)(f)) i s closed, V f e S(X) . So Lemma 4.4 implies that h 1 i s continuous. Similarly, <KH \g)) = h "'"((Kg)) > V g e S(Y) . And so h is a homeomorphism. Theorem 4.5. I f X is Hausdorff and i f V x e X , x has a basis, 5 » of open sets such that G - G is at most a singleton, N/G e S ; then X is an S-space. Proof. I f X is a singleton, nothing to prove, so we assume X has more than one point. Let x e X and x e G e 8^ , and chose any y e X such that x =j= y . Define f : G + X by f = < [ y] ' ^ % ? • f i • Then f is continuous with the desired proper-1 [z] , xf G ~ G = {z} ties, and B is an S-basis. Hence X is an S-space. x If a space has a basis of sets which are both open and closed, then we say that X i s 0-dimensional. Corollary 4.5.1. A 0-dimensional Hausdorff space is an S-space. A space is said to be locally Euclidean i f each point has an open neighbourhood about i t that i s homeomorphic to Euclidean n-space,. E n , for some n > 1 (n depends on the point in question). Theorem 4.6. A locally Euclidean space is an S-space. Proof. F i r s t we note that every locally Euclidean space i s regular, Hausdorff, and a union of homeomorphic images of E n , possibly for many different n's. Also we note that homeomorphic images of S-spaces are again S-spaces. So the proof i s complete with the following two lemmas. Lemma 4.6.1. E n is an S-space, \/n >_ 1 Lemma 4.6.2. If X i s a regular Hausdorff space which i s the union of a collection of open subspaces which are S-spaces, then X i s an S-space. n Proof of Lemma 4.6.1. If x e E and d the usual Euclidean metric, then V e > 0 we define N (x) = {y e E n : d(x,y) < e} . Also l e t x e l denote the i ^ coordinate of x , V i = l,2,...,n . Now define f. : N (x) -> E 1 as follows: f.(y) = y. + d(x,y) - e,\/y e N (x) , I c i i e V i = l,2,...,n . Then each i s continuous, so the function f : N £(x) + E n defined by f(y) = (^(y) ,... ,f n(y)) ,V y e N £(x) , i s continuous, since each projection map is continuous. And f has the property that f(y) = y ,Vy e N e ^ ^ ~ 3 1 1 ( 1 f^ x^ = ^ x ~ e'' *" ' X n _ G ^ ^  Hence neighbourhoods of this type form an S-basis for E n . So E n i s an S-space. 8. Proof of Lemma 4 . 6 . 2 . Let x e X . Let H be an open subset of X which is i t s e l f an S-space containing x . Let B be an S-basis for x x i n H Since X is regular, there exists an open subset V of X and a closed subset F of X such that x e V C F <T H . Let A * B = {G e B : G C V} , then B is a basis for x i n X .We want x x — x to show that i t is an S-basis. Let G e B . If G = {x} , we are done. Otherwise we know that there i s a continuous function r i f : G~->-H. such that f(x) \ x , but f(y) = y ,yy e G^ ~ G . (Note G means the closure of G relative to H , so G = G .) But note n X that G T T C F ^ H , and F i s closed in X , so G" = ~G - G . So ri — H X G i s i n fact an S-neighbourhood of x i n X . So X i s an S-space. Now we try to broaden our knowledge of which spaces are T-admis-sible, and for what T's. So we define two more classes of spaces. 4 * Definition 4 . 7 . 1 . X i s a T -space i f i t i s T^ and for every closed subset, F , of X and \/y e X ~ F , there exists a k^ E T(X) and an x E X such that y A k _ 1({x }) t> F . y * y y ~ Definition 4 . 7 . 2 . X i s a T^-space i f i t i s T^ and for every closed subset, F , of X there exists a k E T(X) such that k_('F) is f i n i t e F F (i.e., k assumes only a f i n i t e number of values on F ) and such that r k^CkpCF)) = F A In [ 6 , p. 2 9 5 , Theorem 1] Magill proves that S -spaces are S-admissible. We generalize this to: 4 The definition of T -space is adapted from Magill's definition of A S -space i n [ 6 ]. The condition of T^-'ness was found by the author. 9. * Theorem 4.8. If T(X) £T(X) , for every T^- and T -space X , * then T - and T^-spaces are both T-admissible classes. Proof. Let F be any closed subset of X and suppose H : T(X) -* T(Y) is an isomorphism. Case i) X and Y are T*-spaces, then -\f y e X ~ F l e t k y e T(X) and x e X be as in definition 4.7.1. Then clearly f\ k "*"({x }) . y . • "~ yeX~F y But Vy e X ~ F ., y \ k " L ( { x }) => (X ~ F) fl ( C\ k ^ x })) =0 . 7 . yeX~F y 7 So F = f j k~ 1({x }) . Hence the class {f - 1({x}) : x e X and yeX~F 7 y f E T(X)} forms a basis for the closed sets of X (each member of this class i s closed since T(X)OT(X)) . Similarly {g"*"({y}) : y e Y and g e T(Y)} i s a basis for the closed sets of Y So now to show h 1 i s continuous we need only show that h(f _ 1({x») i s closed, \/x e X and V f e T(X) . So let f e T(X) and x e X , then H(f) = g <=> hofoh 1 = g => h(f _ 1({x})) = g - 1(h({x})) . But h a bijection => h({x}) i s a singleton, and g e T(x) implies that g 1(h({x})) i s closed. Hence h 1 is 'Continuous. Similarly we get that h i s continuous. Case i i ) X and Y are T -spaces. Let k e T(X) be as i n definition 4.7.2, then i f g = H(k p) we have h°k F°h _ 1 = g => h(F) = Mk^CkpCF) ) ) = g^ChCk ( F ) ) ) . But k„(F) is f i n i t e = > h ( k ( F ) ) is f i n i t e . And . r r r g e T(X) now says in fact that h(F) is closed. So h 1 is continuous. In the same manner h is continuous as well. 10. A It i s easy to see that some T^-spaces are T -spaces, and vice-versa, whether one i s class i s bigger or how largely the classes intersect i s unknown as yet. Below we show some idea of how large the class A of T -spaces i s . Theorem 4.9. Suppose X i s a 0-dimens_ional Hausdorff space and suppose T(X) i s such that whenever X = Al/(X ~ A) , where A is a non-empty, open, and closed subset of X with X - A also non-empty, then for some a e A the function f : X -> X defined by A f ( z) j ' » 2 e X ~ A belongs to T(X) . Then X i s a T -space, z e A Proof. Let F be a closed subset of X and y e X ~ F , then there exists a subset, G , of X which i s both open and closed such that y e G C x ~ F . Then by assumption 3 x e X - G such that the function f e X X defined by f(a) = ( Z ' , ; z i J belongs to T(X) . But now / X , Z £ vj — 1 A y e f ({x}) 3 F . So X is a T -space. In [6, Theorem 2] Magill proves that 0-dimensional Hausdorff A spaces are S -spaces. Using our Theorem 4.9 we generalize this to: Corollary 4.9.1. If X is a 0-dimensional Hausdorff space, then X is a T -space i f S(X) D C(X) C I ( X ) . Proof. Any function f defined as in the statement of Theorem 4.9 i s continuous and closed. We note here that i f T^(X) T£(X) and X A A i s a T^-space, then X i s a T^-space as well. The same holds for T.-spaces as well. So we get the following restatement for theorem 3 of [6] 11. Theorem 4.9.2. I f X i s a completely r e g u l a r Hausdorff space c o n t a i n i n g at l e a s t one non-degenerate path between two d i s t i n c t p o i n t s , and i f S ( X ) t £ T T ( X ) , then X i s a T*-space. Proof.' Let F be a closed subset of X and l e t y e X - F . Then .complete r e g u l a r i t y i m p l i e s t h a t there e x i s t s a continuous f u n c t i o n f : X -> [0,1] = I such that f (F) ,F 0 and f (y) == 1 . By assumption there are two d i s t i n c t p o i n t s x,z e X and a continuous f u n c t i o n g : I -*• X such t h a t g(0) = x and g ( l ) = z Then g o . f e S(X) => g o f e T(X) . But a l s o y £ ( g ° f ) _ 1 ( { x } ) 2 . F A So x i s a T -space. I f X and Y are two t o p o l o g i c a l spaces, then by X U Y we denote the space w i t h a b a s i s of open sets c o n s i s t i n g of those which are open i n e i t h e r X or Y .We assume X and Y are d i s j o i n t . C o r o l l a r y 4.9.3. I f X i s a completely r e g u l a r Hausdorff space then A X i s a subspace of an S - s p a c e ; v i z . , of Y = X U I P r o o f . Y c e r t a i n l y s a t i s f i e s the hypothesis of Theorem 4.9.2. There e x i s t completely r e g u l a r Hausdorff spaces which are not A S -spaces. For example i n [1] Cook gives an example of a compact, m e t r i c , one-dimensional, indecomposable continuum, Z } such t h a t S(Z) c o n s i s t s e n t i r e l y of K(Z) and the i d e n t i t y map. Therefore Z cannot be an A S -space; and we have The f o l l o w i n g d i s c u s s i o n of S -space, hence 4.9.2 to 4.10, i s due to M a g i l l i n [6].. Where p o s s i b l e we make the obvious g e n e r a l i z a t i o n s to A T -space. 12. Corollary 4.9 • 4. : The property of being an S -space is not hereditary. Proof. Let Y = Z U I , then by Corollary 4.9.3 Y is an s"-space with Z as a subspace. Now continuing with Cook's example we get: * Theorem 4.10. There is an S -space which is not an S-space; v i z . , Y - Z UI is such a space. Proof. Let x e Z and G any open subset of Z such that Z - G has more than one point. If Y i s an S-space, then there would exist a continuous function f : G Y such that f (x) •=}= x and f(y) = y , V y E G - G . Define g e Y Y by g(y) = ff(y) , y e G . Then g is [ y , y e Y ~ G •continuous i f f i s . But since Z is -connected, then g(Z) would be connected. But Z - G is not empty, so 3w £ Z ~ G such that g(w) = w •=> g(Z) /"} Z =j= 0 . Then Z connected implies that g(z) C Z => g e S(Z) . But g is neither a constant (since Z ~ G has more than one point) nor the identity (since g(x) ={= x ) , so g ^ S(Z) , which i s a contradiction. So Y is not an S-space, but by Corollary 4.9.3 Y is an S -space. To extend some of the results above we look at a class of semi-6 groups referred to as A-semigroups. Definition 4.11.1. Let A(X) be a family of subsets of X such that X e A(X) ; {x} £ A(X) , \/x e X ; and 0 £ A(X) . Then we say that The following discussion of A-semigroups, hence 4.11.1 through to 4.12.3 are due to Magill in [3]. 13. T(X) i s a A-semigr.o.up i f i) T(X) = {f e X X : f(A) E A(X) , V A E A(X)} and i i ) V A e A (X) , ^ f e T(X) such that f(X) = A Note that K(X) CT(X) because of property i i ) . Lemma 4.11.2. Let T(X) and T(Y) be A-semi-groups and H : T(X) -> T(Y) be an .isomorphism, then i) h(A) e A(Y) , V A e A(X) , and i i ) h _ 1(B) e A(X) ,VB E A(Y) Proof. Let A e A(X) , then 3f e T(X) such that f(X) = A , then h(A) = h(f(X)) = h(f(h _ 1(Y))) = H(f)(Y) e A(Y) . Similarly for part i i ) . Theorem 4.11.3. The class of T^-spaces i s C-admissible. Proof. Let X and Y be T^-spaces and H : C(X) -*• C(Y) be an isomor-phism. Define A(X) = { F C X : F is closed and F =f= 0} . Similarly for A(Y) . Let F e A(X) , then choose any x e F and define f E X^ by f(y) = fy , y e F , then f is clearly closed and (x , y e X ~ F f(X) = F . So C(X) is a A-semigroup. Similarly C(Y) i s a A-semigroup. But by Lemma 4.11.2 both h and h are closed. So h is a homeomorphism. Now we use somewhat the same approach to the space of connected functions. Unfortunately a weaker result must follow. Definition 4.12.1. X i s a U-space i f i t i s connected and for every connected subset A of X , 3f e U(X) such that f(X) = A . 14. One can easily see that not a l l U-spaces are U-admissible. We look at X = { (x,y) : y = sin(l/x),Vx e (0,1] and y = 0 i f 2 x = 0} . Then X , with the induced topology from E , and I , with the usual topology, are U-spaces. But the map h : X -> I defined by h(x,y) = x , V(x,y) e X , i s one-to-one, onto, and biconnected and so induces an isomorphism, H , from U(X) to U(I) given by H(f) = h°f°h - 1, \/f e U(X) . But h" 1 is not continuous at 0 . But, of course, from the same type of argument as in the proof of theorem 4.11.3 we get that i f H : U(X) -* U(Y) is an isomorphism and X and Y are U-spaces, then h is biconnected; i.e., both h and h ^ are connected. The question then i s : when i s such an h a homeomorphism? From Pervin and Levine in [7, Theorem 3.10, p. 495] we get that any biconnected map between two locally connected, compact, Hausdorff spaces is a homeomorphism. Hence: Theorem 4.12.2. Locally connected, compact, Hausdorff U-spaces are a U-admissible class. For usefulness we need some knowledge of the extent of U-spaces. Theorem 4.12.3. If X is a connected, completely regular, Hausdorff space with cardinality c (the cardinality of the continuum), then X is a U-space. Proof. Let x,y be two distinct points i n X . Then complete regu-l a r i t y implies that 3 f : X-> I such that f i s continuous and f(x) = 0 and f(y) =1 . Now since f(X) is connected and contains both 0 and 1 , then f(X) = I . I f a e I , let 15. ° ' a lV . cL ... n (where a. i = 0 or 1) denote the non-terminating binary expansion of a . Define g on I by g(a) = {0 , a = 0 I 1 n lim sup (— I a±), a ={= 0 i=l Then from Kurotowski [2, p. 82] we see that g has the property that i f A is any non-degenerate half-open, open, or closed interval i n I , then g(A) =1 . If B C-1 is connected, then by assumption the cardinality of B is less.than or equal to c , hence there exists some X onto map h : I -»• B • . Thus h°g°f e X . Let D be any connected subset of X , then f(D) is either a singleton or a non-degenerate interval. In the- former case (h°g°f) (D) is-a singleton, hence, -connected,. In the latter case g(f(D)) = I , so (h°gof)(D) = B , which is assumed connected. Hence h°g°f e U ( A ) and (h°g°f)(X) = B. r As an end to this chapter, and as an introduction to the neat, we note: Corollary 4.13. Let X be a topological space, then every automorphism of T(X) is inner i f any of the following hold: i) X i s an S-space,and T(X) = S(X) . i i ) X is either a T -or a T^-space and {h e'X^ : h is a homeomorphism} ^  T(X) ^ r(X) ; i i i ) T(X) i s a A-semigroup; or iv) X is ^ and . T(X) = C(X) . 16. P r o o f . i ) Theorem 4.3; i i ) Theorem 4.8; i i i ) Lemma 4.11.2; and i v ) Theorem 4.11.3. 5. Automorphisms o f T ( X ) . B e f o r e we b e g i n we i n t r o d u c e a few more n o t a t i o n s to f a c i l i t a t e the d i s c u s s i o n . Suppose V i s some subs e m i -group o f X X , t h e n : D e f i n i t i o n 5.1.1. Aut(£>) = the s e t of a l l automorphisms o f V . D e f i n i t i o n 5.1.2. Inn(P) = {H e Aut(P) : H i s i n n e r } . • D e f i n i t i o n 5.1.3. Z(P) = c e n t e r o f V = {f e V : f°g = g°f, Vg e V} . D e f i n i t i o n 5.1.4. B(P) = { b i j e c t i o n s h e V : h " 1 e V} . D e f i n i t i o n 5.1.5. F o r any s e t A , i denotes the i d e n t i t y map on A A i / i . e . , i ^ i s t h a t map i n A such t h a t i ^ ( x ) = x > e A . We always h e r e assume t h a t K ( X ) ^ T(X) Lemma 5.1.6. I f i E T(X) , then Z ( T ( X ) ) = { i v } . P r o o f . C e r t a i n l y we always have i e Z ( T ( X ) ) . Now l e t f e Z ( T ( X ) ) x then fog = gof , \/g £ T(X) => [ f ( x ) ] = f ° [x] = [x]°f = [x] , l / x e X <=> f = i„ . Lemma 5.1.7. Suppose i e T(X) and $ : B( T ( X ) ) -»- I n n ( T ( X ) ) i s ' X d e f i n e d by <J>(h) = H where H ( f ) = hofoh"1, Vf e T(X) . Then <j> i s an i s o m o r p h i s m and Z ( B ( T ( X ) ) ) = { i } X 17. Proof . That ^ i s a homomorphism onto i s t r i v i a l . And the uniqueness par t of Lemma 3.3 t e l l s us that i f $(h^) = ^(h^) , then h^ = ; i . e . , <j> i s one-to-one. Now suppose h e Z(B(T(X))) , then <J>(h)(f) = h o f o h " 1 , \/f E T(X) . But h e Z(B(T(X))) => <j>(h)(f) = h°foh _ 1 = f = i ^ o f o i " 1 , so h = i . • So f o r any T(X) we can descr ibe the inner automorphisms of T(X) . We are here i n t e res t ed i n the case where every automorphism i s i nne r . Coro l l a ry 4.13 y i e l d s many r e su l t s i n th i s regard . We f i r s t concern ourselves with a s p e c i f i c problem. 7 X Theorem 5.2. Let X be the rea l s and V = D(X) = {f e X : f has a ( f i n i t e ) de r i v a t i v e everywhere} , then Aut(1?) = Inn(1?) Proof . Let H E Aut(P) . I t i s c l ea r that X i s a D^-space. For OO example, suppose F i s a c losed subset of X , then X ~ n=0 as a d i s j o i n t union of non-degenerate open i n t e r v a l s . Define k E X X by k(x) = f 0 , x E F 2TTX—IT ( a +b ) ( b n -a n ) [ cos ( _l n ) + 1 ] , x E ( a n , b n ) n n then k e P and k - 1 ( { 0 }) = F . So by Theorem 4.8 we have that h i s a homeomorphism. But being a homeomorphism of the r ea l s we know that h i s s t r i c t l y monotone, hence i s d i f f e r e n t i a b l e somewhere (see Royden [8, p. 96 ] ) , say at x^ . Let X Q E X and def ine 7 This i s Theorem 2.1 of [4] . 18. f(x) = X + x - X . o 1 t(x) = hCx+x^ - hXXj) e X Then f e V and t i s a homeomorphism such that t(0) =0 . So we •get that i f x ^ 0 and g = H(f) = h°f°h 1 , then g(h(x 1) ..+ t ( x ) ) - g(h(x 1)) h( X l+x) - h ( X l ) h(x Q+x) - h ( x Q ) t(x) X X This equation i s seen to be v a l i d i f we merely substitute f o r f(x) and t(x) and note that g°h = h°f . Now since the l i m i t as x -+• 0 on the l e f t side of this equation e x i s t s and equals g'(h(x^)h'(x^) , then the l i m i t on the r i g h t side exists and must equal, by d e f i n i t i o n , h'(x o) But x^ i s a r b i t r a r y , hence h e V . S i m i l a r l y (by considering H 1 ) -1 h * - * From Lemma 5.1.7 we immediately now get: Corollary 5.2.1. Under the assumptions of Theorem 5.2 AutCP) i s i s o -morphic to B(P) = the set of s t r i c t l y monotonic functions i n X^ which have a f i n i t e d e r i v a t i v e everywhere. In [4, Corollary 2.3] M a g i l l proves that i f X i s the r e a l s and D( x) i s the differentiable maps, then every automorphism of D(X) has a unique extension to an inner automorphism of S(X). ¥e can generalize this to: Theorem 5.3. Suppose 8 and C are subsemigroups of X X such that K(X) B ^ C and Aut(B) = Inn(B) , then every automorphism of B has a unique extension to an inner automorphism of C . 19. Proof. Let H e Aut(B) . Define H* : C -> C by H*(f) = h.ofoh 1 , t/f e C • Then H i s an extension of H , because Aut(8) = Inn(B) => h e B(B)'=> h e B(C) => H e Inn(C) . Now, i f K is some other exten-sion of H , then, using Lemma 3.3, we have, x e X K([x]) = [k(x)] = H('[x]) = [h(x)] •=> h = k=> -H* = K . Now to try to extend Theorem 5.2 to say the entire (everywhere analytic) maps on the complex numbers, we immediately run into d i f f i c u l t y : i f V is this semigroup of entire maps and h is the complex conjugate function (i.e., i f z = x + iy, x,y real, then h(z) = x - iy = z ) then H e Aut(P) defined by H(f) = h°foh ^ , r f f e V , is an automor-phism but i s not inner since h £ V Another method of extension has been attempted, and this i s to the semigroup of Frechet-differentiable maps on a real Banach space, X To say a function f e ^ is Frechet-differentiable we mean that there exists a map <$f : X x X ->• X which i s continuous and linear in the second variable and i s such that Va e X we have |jf(a+x) - f(a) -6f(a,x))l _ lim T I— i i = 0 x ->0 l l X i i Then 6f(a,x) i s called the f i r s t derivative of f at a with increment x . The problem, stated by Yamanuro In [9], i s : i f V is the set of a l l Frechet-differentiable maps on X , then i s Aut(P) = Inn(P) ? The answer i s not yet known. The best result so far is that of Yamanuro in [10]: an automorphism H e Aut(P) i s inner i f f i t i s uniform; where uniform means that Ve > 0 and for every sequence of real numbers oo . {a } i such that a t 0 , \J but a -> 0 as n -»• °° , there exists n n=l n 1 y n n 20. a 6 > 0 such that i f ||x|j < <5 , then sup n>l h ( a h - 1(x+h(0))) - h(0) n _ x a n < eMf The proof of this i s long and complicated. If we .try and .look at other semigroups., we find there i s usually very much d i f f i c u l t y in showing whether or not a l l automorphisms are inner, An example which i s not so d i f f i c u l t i s the following. Theorem 5.4. Let X be a set and V a subsemigroup of X^ such that K(X) CZ£> . If there exists a collection of subsets of X , S_ , con-taining a l l f i n i t e subsets of X . , such that i ) V = {f e X X : f ^ C A ) e S_, VA e S) and i i ) V A E S.,3f e V and a f i n i t e subset F of X such that f _ 1 ( F ) = A ; then Aut(£>) = Inn(P) Proof. Let H E Aut(P) and A e S_ . Then by assumption there exists an f e V and a f i n i t e subset F of X such that f "''(F) = A . Let g = H(f) = h o f o h " 1 . Then h(A) = h ( f _ 1 ( F ) ) = g - 1(h(F)) . And since h(F) is f i n i t e , property i) says that h(A) E S_ . Hence h £ V Similarly h E V Examples of where Theorem 5.4 would apply are any measure space where f i n i t e sets are measurable. and S_ i s the collection of measurable sets, or where S_ is the collection of closed sets of some S^-space. 21. 6. C o n c l u s i o n s . Some of the r e s u l t s g i v e n above g i v e c r i t e r i a f o r two spaces b e i n g homeomorphic. For example i f X and Y are T-admissab le and there e x i s t s an H : T(X) -»- T(Y) which i s an i somorphism, then X and Y are homeomorphic. I f A u t ( T ( X ) ) = Inn(T(X)) we can sometimes say something t o o . For i n s t a n c e i f P(X) = the n e a r - r i n g of Fre.chet d r f f e r e n t i a b l e maps, where sum and c o m p o s i t i o n are the n e a r - r i n g o p e r a t i o n s , then i t i s e a s i l y seen tha t a l l r i n g automorphisms of D(X) are u n i f o r m , hence i n n e r . So D(X) and D(Y) are i s o m o r p h i c i f f X and Y are d i f f e o m o r p h i c . I t i s of g e n e r a l mathemat ica l i n t e r e s t to know when a l l the automorphisms of a semigroup are i n n e r . I n t h i s r e g a r d there i s much room f o r r e s e a r c h . For i n s t a n c e , more examples of f a m i l i a r semigroups w i t h t h i s p r o p e r t y c o u l d be searched f o r . A l s o open f o r i n v e s t i g a t i o n i s the q u e s t i o n , i f V i s a semigroup such tha t Aut(P) = Inn(P) , what then can we conclude about the semigroup V i t s e l f ? These q u e s t i o n s f o r r e s e a r c h are i n t e r e s t i n g i n tha t they can be s i m p l y s t a t e d and u n d e r s t o o d , but not so s i m p l y can they be. answered. For example Theorem 5.4 concerns semigroups h a v i n g the p r o p e r t y tha t there i s some c o l l e c t i o n S_ of subsets such t h a t the p r o p e r t y f "^(A) £ S_, )/k e S_ , comple te ly c h a r a c t e r i z e s the elements of the s e m i -group. I f f o r a c e r t a i n semigroup such a c o l l e c t i o n of subsets i s f o u n d , then much about the semigroup may be d e d u c i b l e from s t u d y of the c o l l e c -t i o n of s u b s e t s . I f such a c o l l e c t i o n were found f o r say the Fre 'chet-d i f f e r e n t i a b l e maps, then perhaps the q u e s t i o n of whether a l l automor-phisms are i n n e r or not c o u l d be answered. 22. References 1. Howard Cook, "A continuum which admits only the identity mapping onto a non-degenerate subcontinuum", Abstract 625-3, Amer. Math.  Soc. Notices 12(1965), p. 545. 2. C. Kuratowski, Topologie II (Warsaw, 1950). 3. K.D. Magill, Jr., "Semigroups of functions on a topological space", Proc. London Math. Soc. (3) 16(1966), p. 507-518. 4. , "Automorphisms of the semigroup of a l l differen-tiable functions", Glasgow Math. Journal 8(1967), pp. 63-66. 5. , "Semigroups of continuous functions", Amer. Math. Monthly 71(1964), pp. 984-988. 6. , "Another S-admissible class of spaces", Proc. Amer. Math. Soc. 18(1967), pp. 295-298. 7. W.J. Pervin and N. Levine, "Connected mappings of Hausdorff Spaces", Proc. Amer. Math. Soc. 9(19.5.8)., p.p. 488-496.. 8. H.L. Royden, Real Analysis, 2nd edition, The MacMillan Co., New York (1968). 9. S. Yamamuro, "A note on semigroups of mappings on Banach spaces", Journal Australian Math. Soc. 9(1969), pp. 455-464. 10. , "On the semigroup of differentiable mappings", Journal Australian Math. Soc. 10(1969),. pp. 503-510. 

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