MSC:  54C35  Function spaces in general topology 

54A20  Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 

54D30  Compactness 

46A20  Duality theory for topological vector spaces 

54C25  Embedding 
Abstract:
We start with the paper [1] and thus come back to continuous convergence and to the
characterization of compactness with respect to this convergence structure for the space
C(X, Y ) of continuous functions, where X and Y are topological spaces. More generally
we can use for X, Y convergence spaces, as was done for instance in [11] and [15]. But in
the first paper of this title X and Y were topological spaces and we will continue with this assumption.
...
[1] Bartsch, R., Dencker, P., and Poppe, H. : AscoliArzela Theory based on contin
uous convergence in an (almost) nonHausdorff setting. Categorial Topology, E. Guli,
ed., Kluwer Academic Publisher, Dordrecht 1996, 221 – 240
[11] Mynard, F. : A convergencetheoretic Viewpoint on the ArzelaAscoli theorem. Real.
Anal. Exch. 38, No. 2, 2013, 431 – 444
[15] Poppe, H. : Compactness in General Function Spaces. Deutscher Verlag der Wissenschaften, Berlin 1974