For topological spaces X and Y denote the set of all continuous functions from X to Y by C(X,Y). For a function \(f\in C(X,Y)\) and a family \({\mathcal C}\) of subset of Y set \[ B(f,{\mathcal C})=\{g\in C(X,Y):\quad \{\{f(x),g(x)\}:\quad x\in X\quad refines\quad {\mathcal C}\}\}. \] Then \(\tau =\{U\subseteq C(X,Y):\) for all \(f\in U\) there is an open cover \({\mathcal C}\) of Y with B(f,\({\mathcal C})\subseteq U\}\) is a topology on C(X,Y) called the limitation topology. If “open cover of Y” in the above definition is replaced by “family of open subsets of Y which covers f(X)” one gets the modified limitation topology \(\tau '\). The author shows that \(\{\) B(f,\({\mathcal C}):\) \({\mathcal C}\) is an open cover of \(Y\}\) is a local base of \(\tau\) at \(f\in C(X,Y)\) if Y is paracompact, and consequently a subset \(E\subseteq C(X,Y)\) is \(\tau\)-dense in C(X,Y) if and only if B(f,\({\mathcal C})\cap E\neq \emptyset\) for all f and all \({\mathcal C}\). An analogous result is proven for \(\tau '\). Furthermore, if X is compact, metrizable and Y metrizable then \(\tau\) and \(\tau '\) are equal to the compact-open topology. Two more related topologies are defined and the question of whether they coincide is discussed.