# Topology of uniform convergence

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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)

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
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article