This paper conglomerates our findings on the space C(X) of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the m-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of Z-ideals of C(X) induced by the U I and the m I-topologies on C(X) [5]. Motivated by the definition of the m I-topology, another generalization of the topology of uniform convergence, called U I-topology, is introduced here. Among several other results, it is established that for a convex ideal I in C(X), a necessary and sufficient condition for U I-topology to coincide with m I-topology on C(X) is the boundedness of X Z[I] in X. As opposed to the case of the U I-topologies (and m I-topologies) on C(X), it is proved that each U I-topology (respectively, m I-topology) on C(X) is uniquely determined by the ideal I. In the last section, the denseness of the set of units of C(X) in C U (X) (= C(X) with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space X. Also, the space X turns out to be a weakly P-space if and only if the set of zero divisors (including 0) in C(X) is closed in C U (X). Computing the closure of C P (X) (={ f ∈ C(X) : the support of f ∈ P} where P is an ideal of closed sets in X) in C U (X) and C m (X) (= C(X) with the m-topology), the results cl U C P (X) = C P ∞ (X) (= { f ∈ C(X) : ∀n ∈ N, {x ∈ X : | f (x)| ≥ 1 n } ∈ P}) and cl m C P (X) = { f ∈ C(X) : f. ∈ C P ∞ (X) for each ∈ C(X)} are achieved.