A note on $C_{c}(X)$

Abstract:

When $X$ is a locally convex topological linear space, the function algebra ${C_c}(X)$ (with continuous convergence) can have a closure operator which has infinitely many distinct iterations. The reverse situation is also possible: $X$ can be a locally compact $c$-embedded convergence space whose closure operator has infinitely many distinct iterations, whereas ${C_c}(X)$ is a topological space.