This article is devoted to a study of locally convex topologies on $\mathbf{H}\left(U\right)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and $\mathbf{H}\left(U\right)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on $\mathbf{H}\left(U\right)$:(a) the compact open topology ${\mathbf{I}}_0$,(b) the bornological topology associated with ${\mathbf{I}}_0$,(c) the ported topology of Nachbin ${\mathbf{I}}_{\omega }$,(d) the bornological topology associated with ${\mathbf{I}}_{\omega }$ ; and(e) the ${\mathbf{I}}_{\omega }$ topological of Nachbin.For $U$ balanced we show these topologies are...