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.