On the cardinality of non-isomorphic intermediate rings of C(X)

Abstract

Let \(\sum (X)\) be the collection of subrings of C(X) containing \(C^{*}(X)\), where X is a Tychonoff space. For any \(A(X)\,{\in }\, \sum (X)\) there is associated a subset \(\upsilon _{A}(X)\) of \(\beta X\) which is an A-analogue of the Hewitt real compactification \(\upsilon X\) of X. For any \(A(X)\,{\in }\, \sum (X)\), let [A(X)] be the class of all \(B(X)\,{\in }\, \sum (X)\) such that \(\upsilon _{A}(X)=\upsilon _{B}(X)\). We show that for first countable non compact real compact space X, [A(X)] contains at least \(2^{c}\) many different subalgebras no two of which are isomorphic in Theorem 3.8.