Editing Box topology (section)

===Convergence in the box topology===
Topologies are often best understood by describing how sequences converge. In general, a Cartesian product of a space <math>X</math> with itself over an [[index set|indexing set]] <math>S</math> is precisely the space of functions from <math>S</math> to <math>X</math>'','' denoted <math display="inline">\prod_{s \in S} X = X^S</math>. The product topology yields the topology of [[pointwise convergence]]; sequences of functions converge if and only if they converge at every point of <math>S</math>.

Because the box topology is finer than the product topology, convergence of a sequence in the box topology is a more stringent condition.  Assuming <math>X</math> is Hausdorff, a sequence <math>(f_n)_n</math> of functions in <math>X^S</math> converges in the box topology to a function <math>f\in X^S</math> if and only if it converges pointwise to <math>f</math> and 
there is a finite subset <math>S_0\subset S</math> and there is an <math>N</math> such that for all <math>n>N</math> the sequence <math>(f_n(s))_n</math> in <math>X</math> is constant for all <math>s\in S\setminus S_0</math>.  In other words, the sequence <math>(f_n(s))_n</math> is eventually constant for nearly all <math>s</math> and in a uniform way.<ref>{{cite web|last1=Scott|first1=Brian M.|title=Difference between the behavior of a sequence and a function in product and box topology on same set|url=https://math.stackexchange.com/q/448575|website=math.stackexchange.com}}</ref>