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=14240