Topology of uniform convergence
The topology on the space of mappings from a set
into a uniform space
generated by the uniform structure on
, the base for the entourages of which are the collections of all pairs
such that
for all
and where
runs through a base of entourages for
. The convergence of a directed set
to
in this topology is called uniform convergence of
to
on
. If
is complete, then
is complete in the topology of uniform convergence. If
is a topological space and
is the set of all mappings from
into
that are continuous, then
is closed in
in the topology of uniform convergence; in particular, the limit
of a uniformly-convergent sequence
of continuous mappings on
is a continuous mapping on
.
References
[1] | N. Bourbaki, "General topology" , Elements of mathematics , Springer (1988) (Translated from French) |
[2] | J.L. Kelley, "General topology" , Springer (1975) |
Comments
If is a metric space with the uniform structure defined by the metric, then a basis for the open sets in
is formed by the sets
, and one finds the notion of uniform convergence in the form it is often encountered in e.g. analysis.
References
[a1] | R. Engelking, "General topology" , Heldermann (1989) |
Topology of uniform convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topology_of_uniform_convergence&oldid=48994