View source for 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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "General topology" , ''Elements of mathematics'' , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR></table>

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR></table>