In this paper, we introduce quasicompact-open topology on 𝐾𝐶(𝑋, 𝑌), the set of all functions from 𝑋 to 𝑌, which are continuous on the compact subsets of 𝑋 and compare this topology with the open-cover topology, the uniform topology and 𝑚-topology. Then, we examine me-trizability, completeness and countability properties of the quasicompact-open topology on 𝐾𝐶(𝑋, 𝑌). Also, we obtain similar results for the open-cover topology and 𝑚- topology on 𝐾𝐶(𝑋, 𝑌).