Abstract. We study a particular class of separating nest generated intersection rings on a Tychonoff space X, that we call complete bases. They are characterized by the equality ß(v(X, ^D)) = w(X, e¡>) between their associated Wallman spaces. It is proven that for each separating nest generated intersection ring <$ there exists a unique complete base "D such that v(X, ty) = v(X, tf)). From this result we obtain a necessary and sufficient condition for the existence of a continuous extension to v(X, 60) of a real-valued function over X. Some applications of these results to certain inverse-closed subalgebras of C(X) are given.The word space will refer to Tychonoff spaces. In this paper we consider the Wallman compactification u(X, <5D) and the Wallman realcompactification v(X, fy ) associated with a given base2 on a space X. For definitions and basic results the reader is referred to [1], [9], [10]. We study the bases t hat coincide with the trace on X of all zero-sets in its associated space v(X, "D ). These bases, that we call complete, have interesting properties. They are characterized by the relation ß(v(X, 6D)) = u(X, <5D).3 For each base ô n X there exists a unique complete base <>D such that v(X, ty) = v(X, <3)). The base <3) is the largest base with the above property and the smallest complete base on X containing ^. Frink [4] has shown that the real-valued functions over a space X which may be continuously extended to o¡(X, 6D) are those which are ^-uniformly continuous. In [3] D'Aristotle defined countable ^-uniform continuity and he showed that it is a sufficient but not a necessary condition for the existence of a continuous extension to v(X, ty ) of a real-valued function over X. A necessary and sufficient condition has been obtained by Bentley and Naimpally in [2, Theorem 6]. We give another condition by means of the base 4).
In order to provide examples of noncomplete bases, a general resultReceived by the editors March 16, 1978 and, in revised form, August 11, 1978 AMS (MOS) subject classifications (1970). Primary 54D60, 54D35.Key words and phrases. Nest generated intersection ring, strong delta normal base, complete base, countable intersection property, ß-closure, ß-dense, algebra, a-algebra."The author wishes to thank the referee for his suggestions. 2By a base on a space X is meant a separating nest generated intersection ring on X (A. K. Steiner and E. F. Steiner 3Two extensions 7, and T2 of a space X are said to be equivalent if they are homeomorphic via a map that leaves X pointwise fixed. In this case we write Tx = T2.
