Une Structure Uniforme sur un Espace F(E,F)

TL;DR

This paper introduces a new uniform topology on the space of functions from a topological space to a uniform space, ensuring continuity preservation and convergence properties, with applications to measurability and Borelianity.

Contribution

It defines the V-congergence topology, the coarsest topology preserving continuity and pointwise convergence on function spaces, with criteria for convergence without limits.

Findings

C(E,F) is closed under the V-congergence topology.

The topology preserves measurability and alpha-Borelianity.

Convergence criteria are established without involving limits.

Abstract

Let E be a topological space and F a uniform space. We introduce a new topology (in fact a uniform structure) called the V-congergence on the space of applications from E to F such that C(E,F) is closed for this topology and the restriction of this topology to C(E,F) is equivalent to pointwise convergence. In other words this topology is the coarsest preserving continuity. We give a criterion of convergence for this topology not involving the limit. Among properties preserved are mesurability and alpha-borelianity for a countable ordinal alpha.