Editing Pointwise convergence (section)

==Topology==

{{See also|Characterizations of the category of topological spaces}}

Let <math>Y^X</math> denote the set of all functions from some given set <math>X</math> into some [[topological space]] <math>Y.</math>
As described in the article on [[characterizations of the category of topological spaces]], if certain conditions are met then it is possible to define a unique topology on a set in terms of which [[Net (mathematics)|net]]s do and do not [[Convergent net|converge]]. 
The definition of pointwise convergence meets these conditions and so it induces a [[Topology (structure)|topology]], called '''the {{visible anchor|topology of pointwise convergence}}''', on the set <math>Y^X</math> of all functions of the form <math>X \to Y.</math> 
A net in <math>Y^X</math> converges in this topology if and only if it converges pointwise. 

The topology of pointwise convergence is the same as convergence in the [[product topology]] on the space <math>Y^X,</math> where <math>X</math> is the domain and <math>Y</math> is the codomain. 
Explicitly, if <math>\mathcal{F} \subseteq Y^X</math> is a set of functions from some set <math>X</math> into some topological space <math>Y</math> then the topology of pointwise convergence on <math>\mathcal{F}</math> is equal to the [[subspace topology]] that it inherits from the [[product space]] <math>\prod_{x \in X} Y</math> when <math>\mathcal{F}</math> is identified as a subset of this Cartesian product via the canonical inclusion map  <math>\mathcal{F} \to \prod_{x \in X} Y</math> defined by <math>f \mapsto (f(x))_{x \in X}.</math> 

If the codomain <math>Y</math> is [[Compact set|compact]], then by [[Tychonoff's theorem]], the space <math>Y^X</math> is also compact.