ISSN:
1572-932X
Keywords:
Heine–Borel property
;
densely continuous form
;
densely equicontinuous
;
densely pointwise bounded
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is σ compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008666504767
Permalink
