Editing CW complex (section)

==== The construction, in words ====
The CW complex construction is a straightforward generalization of the following process:
* A 0-''dimensional CW complex'' is just a set of zero or more discrete points (with the [[Discrete space|discrete topology]]).
* A 1-''dimensional CW complex'' is constructed by taking the [[Disjoint union (topology)|disjoint union]] of a 0-dimensional CW complex with one or more copies of the [[unit interval]]. For each copy, there is a map that "[[Gluing (topology)|glues]]" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the [[Quotient space (topology)|quotient space]] defined by these gluing maps.
* In general, an ''n-dimensional CW complex'' is constructed by taking the disjoint union of a ''k''-dimensional CW complex (for some <math>k<n</math>) with one or more copies of the [[Ball (mathematics)|''n''-dimensional ball]]. For each copy, there is a map that "glues" its boundary (the <math>(n-1)</math>-dimensional [[N-sphere|sphere]]) to elements of the <math>k</math>-dimensional complex. The topology of the CW complex is the [[quotient topology]] defined by these gluing maps.
* An ''infinite-dimensional CW complex'' can be constructed by repeating the above process countably many times. Since the topology of the union <math>\cup_k X_k</math> is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits.