René Bartsch
On a nice embedding and the Ascoli theorem
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock
54C35 Function spaces, See also {46Exx, 58D15}
54D30 Compactness
54C05 Continuous maps
54C25 Embedding
54C60 Set-valued maps, See also {26E25, 28B20, 47H04, 58C06}
Abstract: Besides some remarks on hit-and-miss hyperspaces and
(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.
Keywords: topological function spaces, Ascoli theory, set-open topology, compactness, relative compactness, hyperspaces