JP Journal of Geometry and Topology
Volume 4, Issue 1, Pages 81 - 96
(March 2004)
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A COMPACT-OPEN TOPOLOGY FOR COLLECTIONS OF SET-VALUED FUNCTIONS AND SOME OF ITS PROPERTIES
Terrence A. Edwards (USA), James E. Joseph (USA) and Myung H. Kwack (USA)
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Abstract:
It has been
observed by T. Edwards that the compact-open
topology on the family of continuous functions
between topological spaces is the weak topology
induced on the family by a certain collection of
continuous compact-valued functions. In this
paper, several of Edwards’ results are
generalized to the F-open topology of A.
Wilansky, and it is shown that Edwards’
discovery leads to a compact-open topology for
families of set-valued functions between
topological spaces. The collection of set-valued
functions endowed with this compact-open
topology gives rise to an imbedding theorem for
the space of set-valued functions, which in
turn, in conjunction with results found in old
papers of E. Michael, D. Wulbert and R. Smithson,
leads to characterizations of compact subsets of
such families from Hausdorff spaces and
arbitrary spaces to Hausdorff spaces, and
sufficient conditions for members of such
families to be first countable and second
countable.
In addition, several theorems
on the first countability of the space of
compact-valued functions with one of the
compact-open topologies of Smithson are proved.
One such theorem is the following generalization
of a result of R. Arens: If X is a
Hausdorff hemicompact space and Y is a
Hausdorff locally compact second countable
space, then the collection of upper
semicontinuous compact-valued functions a
from X
to Y satisfying a(a–1(F)
Ì
F for each F
Ì
Y
is first countable. |
| Keywords and phrases: set-valued functions, compact-open topology, function spaces, finite topology, upper semicontinuous, lower semicontinuous, continuous set-valued functions, Ascoli theorem. |
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