By Allen Nussbaum

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**Example text**

10 11 X and Y are second countable convergence spaces then Cc(X, Y) is second countable. In particular, Cc(X) is second countable if Xis. If X is a c-embedded convergence space, the converse also holds. Proof Take a countable basis ß of X and a countable basis V of Y. We claim that the family ß v of all finite intersections of the family {T(B, D) : BE ß , D E V} Chapter 1. Convergence spaces 46 is a basis of Cc(X, Y). Take a filter F -+ fo in Cc(X, Y) and let F o be the filter based on F n ß v .

X is called sequentially determined if it is first countable and countably sequentially determined. 6. 14 Let X and Y be convergence spaces, X first countable and Y countably sequentially determined. Then each sequentially continuous mapping f : X ....... Y is continuous. Proof Assume that F ....... x in X. Since X is first countable, one can assume without loss of generality that F has a countable basis (Fn ). , such that < TJ >;2 f(F). We have to show that TJ converges to f(x). For each n E N, there is a kn such that Assume, without loss of generality, that the sequence (k n ) is strictly monotonically increasing.

Subspaces, countable products and countable projective limits of sequentially determined convergence spaces are sequentially determined again. In general, final structures do not preserve (countable) sequential determinedness. 9 (ii) is a strict inductive limit of second countable, hence sequentially determined convergence spaces. 15. Many important topological and convergence concepts have a countable variation or a sequential variation or both. Such is the case with Choquet and compact. We examine such versions of these concepts.