# Topology of uniform convergence

The topology on the space $ {\mathcal F} ( X, Y) $
of mappings from a set $ X $
into a uniform space $ Y $
generated by the uniform structure on $ {\mathcal F} ( X, Y) $,
the base for the entourages of which are the collections of all pairs $ ( f, g) \in {\mathcal F} ( X, Y) \times {\mathcal F} ( X, Y) $
such that $ ( f ( x), g ( x)) \in v $
for all $ x \in X $
and where $ v $
runs through a base of entourages for $ Y $.
The convergence of a directed set $ \{ f _ \alpha \} _ {\alpha \in A } \subset {\mathcal F} ( X, Y) $
to $ f _ {0} \in {\mathcal F} ( X, Y) $
in this topology is called uniform convergence of $ f _ \alpha $
to $ f _ {0} $
on $ X $.
If $ Y $
is complete, then $ {\mathcal F} ( X, Y) $
is complete in the topology of uniform convergence. If $ X $
is a topological space and $ {\mathcal C} ( X, Y) $
is the set of all mappings from $ X $
into $ Y $
that are continuous, then $ {\mathcal C} ( X, Y) $
is closed in $ {\mathcal F} ( X, Y) $
in the topology of uniform convergence; in particular, the limit $ f _ {0} $
of a uniformly-convergent sequence $ f _ {n} $
of continuous mappings on $ X $
is a continuous mapping on $ X $.

#### References

[1] | N. Bourbaki, "General topology" , Elements of mathematics , Springer (1988) (Translated from French) |

[2] | J.L. Kelley, "General topology" , Springer (1975) |

#### Comments

If $ Y $ is a metric space with the uniform structure defined by the metric, then a basis for the open sets in $ {\mathcal F} ( X, Y) $ is formed by the sets $ U ( f, \epsilon ) = \{ {g } : {\rho ( f( x), g( x) ) < \epsilon \textrm{ for all } x \in X } \} $, 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) |

**How to Cite This Entry:**

Topology of uniform convergence.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Topology_of_uniform_convergence&oldid=48994