ON CERTAIN GROUPS OF FUNCTIONS

Let C(X,G) denote the group of continuous functions from a topological space X into a topological group G with the pointwise multiplication and the compact-open topology. We show that there is a natural topology on the collection of normal subgroups Δ(X) of C(X,G) of the Mp={f∈C(X,G):f(p)=e} which is analogous to the hull-kernel topology on the commutative Banach algegra C(X) of all continuous real or complex-valued functions on X. We also investigate homomorphisms between groups C(X,G) and C(Y,G).


i. INTRODUCTION AND NOTATION.
Suppose X is a compact topological space and suppose C(X) is the algebra of J.S. YANG all continuous real or complex-valued functions on X with the usual pointwise operations and the supremum norm.Then C(X) is a regular commutative Banach algebra with identity and X is homeomorphic to the maximal ideal space A(C(X)) of the algebra C(X), where A(C(X)) is endowed with the Gel'fand topology which coin- cides with the hull-kernel topology since C(X) is regular, [3].If X is a topo- logical space and G is a topological group, let C(X,G) be the topological group of all continuous functions from X into G under pointwise multiplication and the compact-open topology.In Section 2 of this paper, we study spaces of normal sub- groups of C(X,G).There is a natural topology, analogous to the hull-kernel topo- fogy in Banach algebra, for the collection of normal subgroups of the form M M (X,G) {f e C(X,G): e} where e is the identity element of G; P P the resulting topological space will be denoted by A(X).We show that, with some mild restriction on X and G, X is homeomorphic to A(X), and that A(X*) is the one- point compactification of A(X), where X* is the one-point compactification of the locally compact space X.Some theorems on homomorphisms and extension of homomor- phisms in C(X,G) are considered in Section 3. We also prove a correct version of a theorem originally stated in [7, theorem 8].
All spaces considered in this paper are assumed to be Hausdorff unless speci- fled.For topological spaces X and Y, the function space F = C(X,Y) is under- stood to be endowed with the compact-open topology whenever it is referred to topologically.10(X,G) or simply I 0 if no confusion should occur, will denote the identity element of the group C(X,G).

THE STRUCTURE SPACES.
For a topological space X and a topolog.ical group G, let P C(X,G).If X is compact and G is a Lie group, then r and M p X, are in general 2 P manifolds (c.f.[i]).It is easy to see that M is locally contractible at I 0, a where a X, if X is a locally compact group locally contractible at a, and that every Mp, p X, is n-simple for every positive integer n if X is a locally com pact contractible space.It is also easy to see that the topological group Y is a group with equal left and right uniformities if so is the group G, and that, if G is the projective limit of the inverse system of topological groups { (G, fss): , 8 A}, then M is the projective limit of the inverse system P {(M_(X,G u) fo p p ): 8 A} where fo p p (f) f f for every f M (X,G).Bc p Throughout this paper the spaces X and G will be subject to the following condition.
DEFINITION i [6].A pair (X,G) of a topological space X and a topological group G is called an S-pair if for each closed subset A of X and x % A, there exists f e F such that f(x) # e and Z(f) {x: f(x) e} n A.
It is clear that (X,G) is an S-pair if X is completely regular and G is path connected or if X is zero-dimensional.It is also clear that X is completely regular if (X,G) is an S-pair, and that X AGa is also an S-pair whenever eA a (X,Gu) is an S-pair for each u e A. Magill called a space X a V-space, [4], if for points p, q, x, and y of X, where p # q, there exists a continuous function f of X into itself such that f(p) x and f(q) = y, and has shown that every completely regular path connected space and every zero-dimensional space is a V- space.It is easy to see that X G) is an S-pair if each (Xa,G), a A, is uA a is an S-pair and if G is a V-space.If G is a topological group such that (G,G) is an S-pair, G may not be a V-space.For example, let G I be the additive group of real numbers with the usual topology and let G 2 be any non-trivial finite group with the discrete topology, then (G I G2, G I G 2) is an S-pair since (GI,GI) and (G2,G2) are S-pairs.Since the topological group G I G 2 is not connected with the identity component isomorphic to GI, G I G 2 is not a V-space as it follows from [4, Theorem 3.5].It is pointed out in [7] that X is hemicompact and G is metrlzable if (X,G) is an S-pair, G is a V-space, and F is first countable.

J.S. YANG
It is well-known (c.f. [2])that, for every topological space X, there exists a completely regular space Y such that C(Y) is (algebraically) isomorphic to C(X), where C(Z) is the ring of continuous real-valued function on the space Z.
Using the similar argument mutatis mutandis as used in the construction of the space Y, it is a straightforward to see that, for every topological space X and a topological group G, there is a completely regular space YG such that C(YG,G) is continuously isomorphic to C(X,G), and that, in the case G is path connected, (YG,G) is an S-pair and the associated space YG is independent of the group G within the category of path connected topological groups.The latter means that YGI YG2 whenever G I and G 2 are path connected groups.It follows from the con- struction of the space YG that X YG if (X,G) is an S-pair.
Because of the remarks Just made above, we shall now assume that (X,G) is an S-pair.
For a collection .of normal subgroups of F C(X,G), we define "*" as follows: If U c and U # , let U* {M : M = nU}, let * .THEONEM i. "*" is a closure operator on if and only if whenever M I and M = M 1 n M2, where M 1 and M 2 are intersections of some subsets of I, then either M M 1 of M = M 2.

PROOF:
It is clear that U* U, (U*)* U*, * , and that U* u V* c (U u V)* for subsets U and V of .Hence "*" is a closure operator if and only if U* u V* = (U u V)* for subsets U and V of 7..Now if sU, and M 2 sV, then (U n V)* {M I: M = M 1 0 M2}.Hence we have the theorem.DEFINITION 2. If "*" is a closure operator on I, we shall refer the result- ing topology, not necessarily Hausdorff, on [ as the S-topology, and the resulting space will be referred to as a G-structure space, or simply structure space, of the space X.
COROLLKRY If [ admits the S-topology, so is every subset of .REMARK 2. If G is path connected, we may speak of structure spaces for the space X without referring to the group since C(X,G) and C(X,R) are isomorphic in this case.LEMMA 3. If a collection of normal subgroup of r admits the S-topology, then a subset A of is closed if and only if there exists a normal subgroup M 0 of r which is the intersection of some subset of such that A {M : M = M0}.In fact, M 0 hA.
PROOF: Suppose A= is closed, then A {M : M nA M0}.
Conversely, suppose that there exists a normal subgroup M 0 of , where M 0 0U for some U c , such that A ={M : M M0}.Then {M : M = hA} A. Hence A is closed.This completes the proof.
If we denote by A(X) the collection of all normal subgroups of F of the form M {f C(X,G): f(p) e}, p X, then the following theorem states that A(X) P admits the S-topology and that the S-topology is Hausdorff if (X,G) is an S-pair.THEOREM 5. A(X) admits the Hausdorff S-topology.
PROOF: Let U and V be subsets of A(X) and let 01 {P X: M U} and p 02 {q X: M V}.It is, by Theorem i, sufficient to show that, if q Mq (POln M) n (k02)' then either Mq pOl Mp and Mq kO2 .Suppose other- wise, then there exist f n M M and g M This implies that PO 1 P q kO 2 q q i and q 2" For if q e i' then there is a net {q} in O I such that q/q.
Then f(q / f(q), and hence f(q) e since f(q e for each s.Similarly, Hence q 01 u 0 2 But (X,G) is an S=pair, let h F such that Z(h) but h(q) e.This would show that h (POln Mp) n (kO 2) but contradiction.Hence either M Mp) or M (k )' and A(X) q P 01 q 02 admits the S-topology.
Next to show that the S-topology is Hausdorff.Let Mp, Mq A(X), where p q. Since X is T2, let 01 and 02 be open sets in X such that P O1, q 02 and 01 02 .If C 2 X O I and C I X 02 then p C I and q C 2. If I 1 and 12 n , then I 1 I 2 A(X) since C 1 u C 2 X, Mp Ii, and kC I kO 2 Mq 12.To see that Mp 12 note that p C2, hence there exists f F such that F(C 2) e but f(p) e.Thus f but f M This shows that M 12 kC 2 P P Similarly, we have M I I.This completes the proof that A(X) is T2, by q Theorem 4.
Note that the S-topology defined above for A(X) is analogous to the hull- kernel topology, which coincides with the Gel'fand topology, on the maximal ideal space of the commutative Banach algebra C(X).
For each I, let A be a closed set of a structure sace .Then, by Lemma 3, there exists a normal subgroup M of F which is the intersection of some subset of such that A {M E ; M M }.If we denoted by [ U M ] the normal subgroup of I" generated by uIM , then we have the following 1 whose proof is straightforward and hence omitted.LEMMA 6. n A {M E [: M [ u M]} THEOREM 7. A structure space I of X is compact if and only if every collec- tion of normal subgroups {N}e_l of r, each of which is the intersection of some subset of Z, such that [ u N ] M for each M Z has a finite subcollection.PROOF: Suppose [ is compact, and let {N } be a collection of normal sub- groups of r, each of which is the intersection of some subset of , such that COROLLARY.A structure space [ of X is compact if every normal subgroup N of F not contained in any element of [ contains a finitely generated normal sub- group of N not contained in any element of . .PROOF: Assume that the stated property holds in [, and let {N } be a collection of normal subgroups of r, each of which is the intersection of some subset of [ such that [ u N ] M for every M in 7..We shall call a normal subgroup N of r free if there is no p X such that f(p) e for each f N.
COROLLARY.A(X) is compact if every free normal subgroup N of F contains a finitely generated free normal subgroup.
THEOREM 8.The mapping : X / A(X) defined by (x) Mx, x X, is a homeomorphlsm.
PROOF: Clearly, is one-to-one and onto.
For the continuity of $, let A c A(X) be closed.Then there exists a normal subgroup M 0 of r such that A {Mx (X): Mx MO}.We shall see that q-l(A) -1 is closed.For this purpose, let {x } be a net in (A) converging to x X.
Then Mx M 0 for each s.If Mx M 0, there exists f M 0 Mx which would imply that f(x) # e, a contradiction since f(x / f(x) and f(x) e for each s.
Next to show that $ is a closed map.Let C be closed in X, and let M 0 n Mx.We claim that $(C) {Mx A(X): Mx = M 0}, which would imply that (C) is closed.It is clear that (C) c {Mx A(X): Mx = MO}.Now let Mx {Mx A(X): Mx = M0}.Then Mx = M 0. Suppose x C, then there exists f C(X,G) such that f(C) e but f(x) #.e.Hence f M 0 but f Mx, a contra- diction.Thus x C, and we have M (C).X If A is a commutative Banach algebra without identity, and if A(e) is the algebra obtained by adjoining an identity to A, then the maximal ideal space z: Z / A(Z) is the mapping of Theorem 8. Then clearly j is continuous and j(x) y if and only if (Mx) M To see that j(x) y if and only if Y (g)(x) g(y) for every g E C(Y,G), let j(x) y.Then (M x) My.Thus ker(hx 4) My.Let g C(Y,G).If g My, hx (g) e, and we have #(g)(x) g(y).If g My, there exists c G such that g C_My, where c__ is the constant mapping of X into c, hence g ck for some k My.Now hx #(g) hx (c_k) hx(c__ #(k)) c(k)(x) c, while g(y) ck(y) c.Hence #(g)(x) g(y) for each g E C(Y,G).PROOF: Note that the group topologies for C(Y,G) and C(X,G) are not rele- vant in the proof of Theorem i0.Hence take discrete topologies for the groups C(Y,G) and C(X,G), then apply the proof of Theorem ii.
As a consequence of the discussions made above, we can now state a correct J.S. YANG version of the theorem originally stated in [7, Theorem 8] in the following.
THEOREM 12.If there exists an isomorphism between groups C(Y,G) and -I C(X,G) which is constant-preserving such that both and # are F-homomorphisms, then X and Y are homeomorphic.
-I PROOF: It is clear that is also constant-preserving if is.Applying Consequently, we have that j(x) x and j (y) y for x X and y Y.
To see this suppose that there exists x X such that % j(x) # x, then we have f C(X,G) such that f( j(x)) # f(x) or f j(x) # f(x).Hence -i -i ( (f) j)(x) # f(x), .andthus (# (f))(x) # f(x) which leads to f(x) # f(x).
Similarly, j (y) y.Hence j is a homeomorphism of X onto Y.
For topological spaces X and Y, it is clear that the space C(X,Y) may be embedded into the space C(X x Z,Y) as a retract for any space Z, and that every homomorphism of the topological group C(X,G) into a topological group L may be extended to a homomorphism of the topological group C(X x Y,G) into L for any topological group L. We shall conclude this paper with the following result concerning an extension of F-homomorphisms.
THEOREM 13.Suppose A is a closed subset of X.Then every constant-pre- serving F-homomorphism h of the topological group C(G,G) into the topological group C(A,G) may be extended to a homomorphism H of the same kind from the topo- logical group C(G,G) into the topological group C(X,G) such that I H h if every continuous function f: A / G may be continuously extended to all of X, where I: C(X,G) / C(A,G) be the map defined by l(f) i being the inclusion map of A into X.
PROOF: For necessity, let f: A G be any continuous function, and let f*: C(G,G) C(A,G) be the natural homomorphism induced by f, namely f*(k) k f for each k C(G,G).Then f* is a constant-preservlng F-homomorphlsm, by Theorem ii.Hence there exists a constant-preservlng F-homomorphlsm H of the topological group C(G,G) into C(X,G) such that I H f*. I(H(id))(a) (I S)(i)(a) f*(id)(a) (i d f)(a) f(a).Hence 8 is an extension of f to all of X.
For sufficiency, assume that every continuous function f: A / G may be extended continuously to all of X, and let h: C(G,G) / C(A,G) be a constant-pre- serving homomorphism of the topological group C(G,G) into the topological group C(A,G).Then there exists f C(A,G) such that h(k) k f for every k C(G,G).
If we denote by f the extension of to all of X, define a function H: C(G,G) / C(X,G) by H(k) k for each k C(G,G).Then H is a constant-pre- serving F-homomorphism and I H h.This completes the proof.
In particular, if X is a normal space, and A a closed subset of X, then every constant-preserving F-homomorphism h of the topological group C(R,R) into the topological group C(Z,R) may be extended to a homomorphism H of the same kind from the topological group C(R,R) into the topological group C(X,G) such that I H=h.

THEOREM 4 . 2 .
If a collection of normal subgroups of r admits the S-topology, then is Hausdorff if and only if for , M 2 I, M 2, there are I 1 and 12, where I I nUl, 12 U 2 and Ul, U 2 = , such that M I Ii, M 2 12, M I 12, M 2 Ii, and I I 12 .PROOF: Suppose that is Hausdorff, and let MI, M 2 , M I # M 2. Thus there are dlsoint open sets U I and U 2 in such that M I c UI, and M 2 I U 2, A 2 I U l, then A I and A 2 are closed and AI, M 2 A m Using eemma 3, we have A i {M : M A.},I i 1,2.If we let I i Ai, i 1,2, then = I I, M 2 = 12 M I 12 M 2 I I and I I 12 n I. Conversely, assume that the stated property holds, and let MI, M l = II, M 2 = 12, M I 12, M 2 Ii, and I I 12 .Let B {M : M ll}, i 1,2.Then B i are closed by Lmma 3, M I BI, M 2 B2, To see that V 1 n V 2 , it suffices to show that if M then either M B I or M B 2. Now M implies M = I I 12 This means that either M = I I or M = 12 since [ admits the S-topology.Hence M B I or M B 2.
, for each e I, let A {M : M m N } then A is closed in , Lemma 3, and 0 A M : M o N ) $. Hence, by h compactness of Z, there exist im u N ]} .Hence u N ] M n i=l for ch M e [.Conversely suppose that [ has the stated property and let {A be a a sI collection of closed sets with the finite intersection property, where A {M [: M N and N is the intersection of some subset of [.Suppose that A .Then {M [: M [ u N ] $, hence [ u N ] W M for each M [.This would imply tt A $, a contradiction.Hence [ is c ompac t.
Let N [ u N ].Then N M for every M in [, thus N contains a finitely generated normal subgroup B such that B M for each M [.Let B [asl a

C
THEOREM ii.A continuous homomorphism of C(Y,G) into C(X,G) is a con- stant-preserving F-homomorphism if and only if there exists f is clear that a homomorphism of the form (k) k f for every k e C(Y,G) is a constant-preserving F-homomorphism.Conversely, if is a constant-preserving F-homomorphism, and if j is the continuous map of X into Y as defined in Theorem i0, then, for each kC(Y,G), (k)(x) k e C(Y,G).COROLLARY.A homomorphism of C(Y,G) ifo X(X,G) is a constant-preservingF-homomorphism if and only if there exists f C(X,Y) such that #(k) )(a) H(id)(i(a)) H(i d) i(a)