New Topologies on the Rings of Continuous Functions

Two new topologies are defined 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 Co(X) and Co?1 (X). The relations between X, Co(X) and Co?1 (X) are studied and closedness of z-ideals and maximal ideals are investigated in Co(X) and Co?1 (X).