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

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.