Topological continuous convergence

Let X be a limit space, Y a topological space. We show that c(X,Y), the limitierung of continuous convergence on LIM(X,Y), is topological whenever X is basic locally compact. For regular Y, local compactness of X is sufficient. In both cases, c(X,Y) coincides with the compact-open topology. If X satisfies a certain regularity condition, the fact that c(X,Y) is topological implies, conversely, that X is (basic) locally compact.The author would like to thank S. Weck for some inspiring discussions.