**MSC:**- 54C35 Function spaces, See also {46Exx, 58D15}
- 54D30 Compactness
- 54C05 Continuous maps
- 54C25 Embedding
- 54C60 Set-valued maps, See also {26E25, 28B20, 47H04, 58C06}

(relative) compactness of unions of

(relative) compact sets, and using these together with a natural

map from a function space between topological spaces into a

function space between their Vietoris hyperspaces, we derive

Ascoli-like theorems for some set-open topologies, nearly

related to the compact-open, and mainly for the compact-open

topology itself. Some Ascoli-like statements are given, where all

requirements are focused on the set of functions, whose (relative)

compactness is in question, not on the base spaces.