Vol. 39, No. 1, 1971

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Uniform convergence for multifunctions

Raymond Earl Smithson

Vol. 39 (1971), No. 1, 253–259

Let be a collection of multivalued functions on a topological space into uniform space. The topology of uniform convergence is defined on , 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 . It is also shown that if 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) is closed in 𝒞, (ii) [x] has compact closure for each x and (iii) is equicontinuous.

Mathematical Subject Classification 2000
Primary: 54C35
Secondary: 54C60
Received: 15 April 1969
Published: 1 October 1971
Raymond Earl Smithson