CW Complex - Properties

Properties

• CW-complexes are locally contractible.
• 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.
• 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 finer than the product topology if neither X nor Y is 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 spaces 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). Note that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compact-open topology; the above statements remain true.
• A covering space of a CW complex is also a CW complex.
• CW-complexes are paracompact. Finite CW-complexes are compact. A compact subspace of a CW-complex is always contained in a finite subcomplex.