Let ℱ be a collection of
multivalued functions on a topological space into uniform space. The topology of
uniform convergence is defined on ℱ, and it is shown that for point compact
functions this topology is larger than the pointwise topology. Some results
are given on uniform convergence of nets in ℱ. It is also shown that if ℱ
consists of point compact continuous functions on a compact space, then the
compact open topology and topology of uniform convergence are the same.
Finally the following Ascoli theorem for multifunctions is obtained. Theorem:
Let 𝒞 be the set of point compact, continuous multifunctions on a compact
regular space into a T2-uniform space. Then ℱ ⊂𝒞 is compact if and only if
(i) ℱ is closed in 𝒞, (ii) ℱ[x] has compact closure for each x and (iii) ℱ is
equicontinuous.
