Topologies for function spaces

A note on locally v-bounded spaces

“…A topology t on C(Y, Z) is called admissible if for every space X, the continuity of a map G : X → C t (Y, Z) implies that of the map G : X × Y → Z. Equivalently, a topology t on C(Y, Z) is admissible if the evaluation map e : C t (Y, Z) × Y → Z defined by relation e(f, y) = f (y), (f, y) ∈ C(Y, Z) × Y , is continuous (see [1]). …”

Completeness properties of the pseudocompact-open topology on C(X)

“…F o x , [12] and soon after, it was developed by R i c h a r d F . A r e n s in [3] and by A r e n s and J a m e s D u g u n d j i in [4]. This topology was shown in [17] to be the proper setting to study sequences of functions which converge uniformly on compact subsets.…”

n-Tuple relations and topologies on function spaces

“…A topology t on C(Y, Z) is called jointly continuous if for every space X, the continuity of a map G : X → C t (Y, Z) implies that of the map G : X × Y → Z (see [5], [1], [2] and [3]). …”

“…(1) The pointwise topology, the compact open topology and the Isbell topology on C(Y, Z) are always splitting (see, for example, [1], [2], [3], [5], [9] and [10]). …”