For a Fréchet algebra A, the continuous spectrum \(S_ c(A)\) is the set of all continuous complex-valued algebra homomorphisms on A endowed with the Gelfand topology, i.e. the topology inherited from the product topology on the space \({\mathbb{C}}^ A\) of all maps \(A\to {\mathbb{C}}\). We say \(S_ c(A)\) is compactly generated if every subset intersecting each compact set in a closed set is itself closed. In this paper, the author gives an example of a complex space (X,\({\mathcal O})\) whose continuous spectrum \(S_ c({\mathcal O}(X))\) is not compactly generated. It follows that the Gelfand topology on the spectrum can be different from the strong topology as well as from the direct limit topology.