Editing Sequence (section)

===Topology===
Sequences play an important role in topology, especially in the study of [[metric spaces]]. For instance:
* A [[metric space]] is [[compact space|compact]] exactly when it is [[sequential compactness|sequentially compact]].
* A function from a metric space to another metric space is [[continuous function|continuous]] exactly when it takes convergent sequences to convergent sequences.
* A metric space is a [[connected space]] if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.
* A [[topological space]] is [[separable space|separable]] exactly when there is a dense sequence of points.

Sequences can be generalized to [[Net (mathematics)|nets]] or [[Filter (set theory)|filters]]. These generalizations allow one to extend some of the above theorems to spaces without metrics.

====Product topology====
The [[product topology|topological product]] of a sequence of topological spaces is the [[cartesian product]] of those spaces, equipped with a [[natural topology]] called the [[product topology]].

More formally, given a sequence of spaces <math>(X_i)_{i\in\mathbb N}</math>, the product space

:<math>X := \prod_{i\in\mathbb N} X_i, </math>

is defined as the set of all sequences <math>(x_i)_{i\in\mathbb N}</math> such that for each ''i'', <math>x_i</math> is an element of <math>X_i</math>. The '''[[projection (set theory)|canonical projections]]''' are the maps ''p<sub>i</sub>'' : ''X'' &rarr; ''X<sub>i</sub>'' defined by the equation <math>p_i((x_j)_{j\in\mathbb N}) = x_i</math>. Then the '''product topology''' on ''X'' is defined to be the [[coarsest topology]] (i.e. the topology with the fewest open sets) for which all the projections ''p<sub>i</sub>'' are [[continuous (topology)|continuous]]. The product topology is sometimes called the '''Tychonoff topology'''.