On the maximal ring of quotients of $C\left( X \right)$

Anthony W. Hager
1966 Bulletin of the American Mathematical Society  
1. Let QiX) denote the maximal ring of quotients (in the sense of Johnson [4] and Utumi [5] ) of the ring C(X) of continuous realvalued functions on the completely regular Hausdorff space X, This ring has been studied by Fine, Gillman, and Lambek [l] and realized by them as the direct limit of the subrings C(V), Va, dense open subset of X (i.e., the union of these C(F)'s, modulo the obvious equivalence relation). From this representation of Q(X), it follows that if X and F have homeomorphic
more » ... ve homeomorphic dense open subsets, then Q(X) and Q(Y) are isomorphic. The full converse to this is false (see below). In this note a proof of the following is described. THEOREM 1. Let X and Y be separable metric spaces. If Q(X) and Q(Y) are isomorphic, then X and Y have homeomorphic dense open subsets. In particular, the spaces R n , w = l, 2, • • • (i£ = the reals) have pairwise nonisomorphic Q's, thus settling a question 1 raised in [l]. That Q(R) is not isomorphic to <2(i£ w ), f°r w>l, was shown by F. Rothberger and J. Fortin. (See [2], and [l, p. 16].) The main purpose of this note is to present a fairly simple solution to this question, and therefore the possible generalizations of Theorem 1 will not be discussed here. These generalizations, and related questions, will be treated in detail in a later paper. The proof of Theorem 1 will now be described.
doi:10.1090/s0002-9904-1966-11585-9 fatcat:xdkoj5lnmbg6ta2odge6er2w4u