On Cardinality Of Non Isomorphic Intermediate Rings Of C(X)

Let $\sum (X)$ be the collection of subalgebras 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 have shown 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.