A topological view of algebraic structures in $C_p(X)$

This manuscript focuses on the topological behaviour of algebraic structures in $C_p(X)$. Closure of ideals in $C_p(X)$ is determined and are used to characterise various topological properties of the underlying Tychonoff space $X$. These include compact space, locally compact space, locally pseudocompact space, $P$-space, almost $P$-space and normal space. Moreover, it has been established that a space $X$ is totally separated space if and only if the set of all units in $C(X)$ is dense in $C_p(X)$. A subring of $C(X)$ is found to be dense in $C_p(X)$ when and only when it separates points.