This paper investigates the compact-open topology on the set of C_k(X) of continuous real-valued functions defined on a Tychonoff space X. More precisely, we study the following problem: If P is a topological property, does there exist a topological property Q so that C_k(X) has P if and only if X has Q? Characterizations of many properties are obtained throughout the thesis, sometimes modulo some “mild” restrictions on the space X. The main properties involved are summarized in a diagram in the introduction. / Ph. D.