Editing CW complex (section)

== Examples ==

=== 0-dimensional CW complexes ===
Every [[Discrete space|discrete topological space]] is a 0-dimensional CW complex.

=== 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>.

=== Finite-dimensional CW complexes ===
Some examples of finite-dimensional CW complexes are:<ref name=":1" />
* '''An [[n-sphere|''n''-dimensional sphere]]'''. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell <math>D^{n}</math> is attached by the constant mapping from its boundary <math>S^{n-1}</math> to the single 0-cell. An alternative cell decomposition has one (''n''-1)-dimensional sphere (the "[[equator]]") and two ''n''-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere").  Inductively, this gives <math>S^n</math> a CW decomposition with two cells in every dimension k such that <math>0 \leq k \leq n</math>.
* '''The ''n''-dimensional real [[projective space]].''' It admits a CW structure with one cell in each dimension.
* The terminology for a generic 2-dimensional CW complex is a '''shadow'''.<ref>{{cite book |last=Turaev |first=V. G. |title=Quantum invariants of knots and 3-manifolds |date=1994 |publisher=Walter de Gruyter & Co. |isbn=9783110435221 |series=De Gruyter Studies in Mathematics |volume=18 |location=Berlin}}</ref>
* A [[polyhedron]] is naturally a CW complex.
*[[Grassmannian]] manifolds admit a CW structure called '''Schubert cells'''.
*[[Differentiable manifold]]s, algebraic and projective [[algebraic variety|varieties]] have the [[homotopy type]] of CW complexes.
* The [[Alexandroff extension|one-point compactification]] of a cusped [[hyperbolic manifold]] has a canonical CW decomposition with only one 0-cell (the compactification point) called the '''Epstein–Penner Decomposition'''. Such cell decompositions are frequently called '''ideal polyhedral decompositions''' and are used in popular computer software, such as [[SnapPea]].

=== Infinite-dimensional CW complexes ===

* The infinite dimensional sphere <math>S^\infty:=\mathrm{colim}_{n\to\infty}S^n</math>. It admits a CW-structure with 2 cells in each dimension which are assembled in a way such that the <math>n</math>-skeleton is precisely given by the <math>n</math>-sphere.
* The infinite dimensional projective spaces <math>\mathbb{RP}^\infty</math>, <math>\mathbb{CP}^\infty</math> and <math>\mathbb{HP}^\infty</math>. <math>\mathbb{RP}^\infty</math> has one cell in every dimension, <math>\mathbb{CP}^\infty</math>, has one cell in every even dimension and <math>\mathbb{HP}^\infty</math> has one cell in every dimension divisible by 4. The respective skeletons are then given by <math>\mathbb{RP}^n</math>, <math>\mathbb{CP}^n</math> (2n-skeleton) and <math>\mathbb{HP}^n</math> (4n-skeleton).

=== Non CW-complexes ===

* An infinite-dimensional [[Hilbert space]] is not a CW complex: it is a [[Baire space]] and therefore cannot be written as a countable union of ''n''-skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
* The [[hedgehog space]] <math>\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subseteq \mathbb C</math> is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is not [[Contractible space#Locally contractible spaces|locally contractible]].
* The [[Hawaiian earring]] has no CW decomposition, because it is not locally contractible at origin. It is also not homotopy equivalent to a CW complex, because it has no good open cover.