In this thesis, we defined two new topologies on C(X). These topologies make C(X) to be a zero-dimensional (completely regular) Hausdorff space. C(X) endowed by these topologies is denoted by C_o(X) and C_(o^(-1) )(X). The relations between X, C_o(X) and C_(o^(-1) )(X) are studied and closedness of z-ideals and maximal ideals are investigated in C_o(X) and C_(o^(-1) )(X).