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H = T* - are precisely the replete spaces. As a special case, one sees that compact spaces have this property. e. 5-4)). 5-2) fact that homeomorphic spaces S continuous functions was noted. 5-l), yet C(UT,I_I) is always the same as C(T,F$. Still, there is a result going in the converse direction. If C(S,R) is isomorphic to C(T,R), then the characters of the algebras are also identifiable, hence so are vS and UT - at least in the sense that they are in 1-1 correspondence. But this is rather weak.
For T with respect to which each x c is the weakest uniform structure C(T,R) is uniformly continuous). (b) UT consists of those points p € BT such that for all sequences (Vn) of neighborhoods of p in BT, (nVn)n T # @. Thus p $' UT if and only if there is some G6 set (in BT) containing p which does not meet T. Proof and ? e. the initial uniformity determined on UT by the functions for x € C(T,F$. We show that UT is the C-completion of T. -complete. ) To show that UT is C-complete, it suffices (see for example Bourbaki 1966a.
ALGEBRAS O F CONTINUOUS F U N C T I O N S Let e denote the identity of C(T,R), the mapping sending each t c T into 1. To see that the map p + p* is onto H, let h be any character. There exists p c BT such that ker h = p , whence h Q uT, for then ker h = ker p* = M consider any x c C(T,F$ show that x B (p) F. P M P and it remains only to show that = p*. To see that p c uT, and its continuous extension xB c C(BT,FU(m]); As x-h(x)e c ker h = we M P p c cl z(x-h(x)e). B Hence there must be a net (t ) from (z(x-h(x)e) converging to p.