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

Let $\sum (X)$ be the collection of subalgebras of $C(X)$ containing $C^{\ast }(X)$, 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$[A(X)]$betheclassofall$B(X)\in \sum(X)$suchthat$\upsilon_{A}(X)=\upsilon_{B}(X)$. 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.