Editing CW complex (section)

== Properties ==
* CW complexes are locally contractible.<ref>{{Cite book |last=Hatcher |first=Allen |title=Algebraic topology |publisher=[[Cambridge University Press]] |year=2002 |isbn=0-521-79540-0 |pages=522}} Proposition A.4</ref>
* If a space is [[homotopy equivalent]] to a CW complex, then it has a good open cover.<ref>{{Cite journal |last=Milnor |first=John |date=February 1959 |title=On Spaces Having the Homotopy Type of a CW-Complex |url=http://dx.doi.org/10.2307/1993204 |journal=Transactions of the American Mathematical Society |volume=90 |issue=2 |pages=272–280 |doi=10.2307/1993204 |jstor=1993204 |issn=0002-9947|url-access=subscription }}</ref> A good open cover is an open cover, such that every nonempty finite intersection is contractible.
* CW complexes are [[paracompact]]. Finite CW complexes are [[compact space|compact]]. A compact subspace of a CW complex is always contained in a finite subcomplex.<ref>[[Allen Hatcher|Hatcher, Allen]], ''Algebraic topology'', Cambridge University Press (2002). {{ISBN|0-521-79540-0}}. A free electronic version is available on the [http://pi.math.cornell.edu/~hatcher/ author's homepage]</ref><ref>[[Allen Hatcher|Hatcher, Allen]], ''Vector bundles and K-theory'', preliminary version available on the [http://pi.math.cornell.edu/~hatcher/ author's homepage]</ref>
* CW complexes satisfy the [[Whitehead theorem]]: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups.
* A [[covering space]] of a CW complex is also a CW complex.<ref>{{Cite book |last=Hatcher |first=Allen |title=Algebraic topology |publisher=[[Cambridge University Press]] |year=2002 |isbn=0-521-79540-0 |pages=529}} Exercise 1</ref>
* The product of two CW complexes can be made into a CW complex.  Specifically, if ''X'' and ''Y'' are CW complexes, then one can form a CW complex ''X'' × ''Y'' in which each cell is a product of a cell in ''X'' and a cell in ''Y'', endowed with the [[weak topology]]. The underlying set of ''X'' × ''Y'' is then the [[Cartesian product]] of ''X'' and ''Y'', as expected. In addition, the weak topology on this set often agrees with the more familiar [[product topology]] on ''X'' × ''Y'', for example if either ''X'' or ''Y'' is finite.  However, the weak topology can be [[comparison of topologies|finer]] than the product topology, for example if neither ''X'' nor ''Y'' is [[locally compact space|locally compact]].  In this unfavorable case, the product ''X'' × ''Y'' in the product topology is ''not'' a CW complex. On the other hand, the product of ''X'' and ''Y'' in the category of [[compactly generated space]]s agrees with the weak topology and therefore defines a CW complex.
* Let ''X'' and ''Y'' be CW complexes.  Then the [[function spaces]] Hom(''X'',''Y'') (with the [[compact-open topology]]) are ''not'' CW complexes in general. If ''X'' is finite then Hom(''X'',''Y'') is homotopy equivalent to a CW complex by a theorem of [[John Milnor]] (1959).<ref name="milnor">{{cite journal |last1=Milnor |first1=John |author-link=John Milnor |year=1959 |title=On spaces having the homotopy type of a CW-complex |journal=Trans. Amer. Math. Soc. |volume=90 |issue=2 |pages=272–280 |doi=10.1090/s0002-9947-1959-0100267-4 |jstor=1993204 |doi-access=free}}</ref>  Note that ''X'' and ''Y'' are [[compactly generated Hausdorff space]]s, so Hom(''X'',''Y'') is often taken with the [[compactly generated space|compactly generated]] variant of the compact-open topology; the above statements remain true.<ref>{{cite web |title=Compactly Generated Spaces |url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |access-date=2012-08-26 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |url-status=dead }}</ref>
* [[Cellular approximation theorem]]