The open-point and bi-point-open topologies on C(X)C(X)

In the definition of a set-open topology on C(X)C(X), the set of all real-valued continuous functions on a Tychonoff space X, we use a certain family of subsets of X   and open subsets of RR. But instead of using this traditional way to define topologies on C(X)C(X), in this paper, we adopt a different approach to define two interesting topologies on C(X)C(X). We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies.