Editing CW complex (section)

=== 1-dimensional CW complexes ===
Some examples of 1-dimensional CW complexes are:<ref name=":1">Archived at [https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20201211210326/https://www.youtube.com/watch?v=HjiooyBH6es&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web |last=channel |first=Animated Math |date=2020 |title=1.3 Introduction to Algebraic Topology. Examples of CW Complexes. |url=https://www.youtube.com/watch?v=HjiooyBH6es&t=25s |website=Youtube}}{{cbignore}}</ref>

* '''An interval'''. It can be constructed from two points (''x'' and ''y''), and the 1-dimensional ball ''B'' (an interval), such that one endpoint of ''B'' is glued to ''x'' and the other is glued to ''y''. The two points ''x'' and ''y'' are the 0-cells; the interior of ''B'' is the 1-cell.   Alternatively, it can be constructed just from a single interval, with no 0-cells.
* '''A circle'''. It can be constructed from a single point ''x'' and the 1-dimensional ball ''B'', such that ''both'' endpoints of ''B'' are glued to ''x''. Alternatively, it can be constructed from two points ''x'' and ''y'' and two 1-dimensional balls ''A'' and ''B'', such that the endpoints of ''A'' are glued to ''x'' and ''y'', and the endpoints of ''B'' are glued to ''x'' and ''y'' too.
* '''A graph.''' Given a [[multigraph|graph]], a 1-dimensional CW complex can be constructed in which the 0-cells are the vertices and the 1-cells are the edges of the graph. The endpoints of each edge are identified with the incident vertices to it. This realization of a combinatorial graph as a topological space is sometimes called a '''topological graph'''.
**[[Trivalent graph|3-regular graph]]s can be considered as ''[[Generic property|generic]]'' 1-dimensional CW complexes.  Specifically, if ''X'' is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from a [[discrete two-point space|two-point space]] to ''X'', <math>f : \{0,1\} \to X</math>.  This map can be perturbed to be disjoint from the 0-skeleton of ''X'' if and only if <math>f(0)</math> and <math>f(1)</math> are not 0-valence vertices of ''X''.
* The ''standard CW structure'' on the real numbers has as 0-skeleton the integers <math>\mathbb Z</math> and as 1-cells the intervals <math>\{ [n,n+1] : n \in \mathbb Z\}</math>. Similarly, the standard CW structure on <math>\mathbb R^n</math> has cubical cells that are products of the 0 and 1-cells from <math>\mathbb R</math>. This is the standard ''[[Integer lattice|cubic lattice]]'' cell structure on <math>\mathbb R^n</math>.