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)