On closed extensions of functions.
G. T. Whyburn has shown that every mapping (by a mapping we will always mean a continuous function) of a topological space into a Hausdorff space is the restriction of a compact mapping with the same range (Trans. Am. Math. Soc., 74, 344-350 (1953)). H. Bauer has proved that every mapping between locally compact Hausdorff spaces can be extended in a topologically unique manner to a compact mapping on a locally compact Hausdorff space with the same range (Math. Ann., 138, 398-427 (1959)). When the range and domain of a mapping are locally compact Hausdorff spaces, Whyburn's and Bauer's extensions are topologically equivalent. It is well known that not every compact mapping is a closed mapping; however, Whyburn has shown that every closed function with compact point inverses is a compact function (these PROCEEDINGS, 54, 688-692 (1965)). The principal aim of this paper is to prove that every function (respectively, mapping) from one topological space to another can be extended in a topologically unique manner to a closed and compact function (respectively, mapping) with the same range.
Dickman
R F
RF
UNIVERSITY OF MIAMI.