A generalization of the strict topology

Robin Giles
1971 Transactions of the American Mathematical Society  
The strict topology ß on the space C(X) of bounded real-valued continuous functions on a topological space X was defined, for locally compact X, by Buck (Michigan Math. J. 5 (1958), 95-104). Among other things he showed that (a) C(X) is /3-complete, (b) the dual of C(X) under the strict topology is the space of all finite signed regular Borel measures on X, and (c) a Stone-Weierstrass theorem holds for /¡-closed subalgebras of C(X). In this paper the definition of the strict topology is
more » ... topology is generalized to cover the case of an arbitrary topological space and these results are established under the following conditions on X: for (a) X is a Ar-space; for (b) X is completely regular; for (c) Ais unrestricted. B0(X) contains (a) the characteristic function y(A) of each compact set K<= X; (b) every function =2™=i anx(Kn), where <*n = 0 for all n, an -> 0, and the sets Kn are compact and disjoint ; and so, in particular, (c) the function 4> = 2"= i «nX({*n})> where (x") is any sequence of distinct points. We shall need the following lemma : Lemma 1.1. If fis a real-valued function on X and ftp is bounded for every in BQ(X) then fis bounded. Proof. Suppose/is not bounded. Choose a sequence (xn) in AT with |/(xn)| -»• oo and put fi e B0(X) butfiji is not bounded. The strict topology on C(X) was defined for locally compact X by Buck [1] by means of a set of seminorms determined by the elements of C0(X). If X is completely regular but not locally compact, C0(X) may be very small (for instance, if X is the rationals [3, p. 109]) and does not yield a useful topology for C(X). We claim that in this case the natural generalization of the strict topology is obtained by letting the role of C0(X) be played by B0(X); the change makes no difference if
doi:10.1090/s0002-9947-1971-0282206-4 fatcat:5prxqkkl4vh7hfw5sqqz65t4ti