A CW-complex is a nice topological space which is or can be built up inductively, by a process of attaching n-disks along their boundary (n-1)-spheres for all : a cell complex built from the basic topological cells .
Being, therefore, essentially combinatorial objects, CW complexes are the principal objects of interest in algebraic topology; in fact, most spaces of interest to algebraic topologists are homotopy equivalent to CW-complexes. Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex.
Also, CW complexes are the cofibrant objects in the classical model structure on topological spaces. This means in particular that every topological space is weakly homotopy equivalent to a CW-complex (but need not be strongly homotopy equivalent to one). See also at CW-approximation. Since every topological space is a fibrant object in this model category structure, this means that the full subcategory of Top on the CW-complexes is a category of “homotopically very good representatives” of homotopy types. See at homotopy theory and homotopy hypothesis for more on this.
W = “weak topology”: Since a CW-complex is a colimit in Top over its cells, and as such equipped with the final topology of the cell inclusion maps, a subset of a CW-complex is open or closed precisely if its restriction to (the closure of) each cell is open or closed, respectively.
A CW-complex is a topological space equipped with a sequence of spaces and continuous maps
and a cocone making into its colimit (in Top, or else in the category of compactly generated spaces or one of many other nice categories of spaces) where each space (called the -skeleton of ) is the result of attaching copies of the -disk along their boundaries to . Specifically, is the empty space, and each is a pushout in Top of a diagram of the form
where is some index set, each is the boundary inclusion of a copy of , and is a continuous map, often called an attaching map. The coprojections of these pushouts give the arrows on which diagram (1) is based.
A relative CW-complex is defined as above, except is allowed to be any space.
A finite CW-complex is one which admits a presentation in which there are only finitely many attaching maps, and similarly a countable CW-complex is one which admits a presentation with countably many attaching maps.
Formally this means that (relative) CW-complexes are special (relative) cell complexes with respect to the generating cofibrations in the standard Quillen model structure on topological spaces: they are those cell complexes which are obtained from a countable transfinite composition of cell attachments, and where in addition the stage is obtained from by attaching cells of dimension , instead of cells of arbitrary dimension.
A cellular map between CW-complexes and is a continuous function such that .
If is an inclusion of CW-complexes, then the quotient is naturally itself a CW-complex, such that the quotient map is cellular.
For example the suspension of a CW-complex itself carries the structure of a CW-complex.
Similarly for pointed CW-complexes: the smash product of a pointed CW-complex with a finite pointed CW-complex is a pointed CW-complex. For example the reduced suspension of a pointed CW-complex is itself a CW-complex.
For and CW-complexes with attaching maps and , then the k-ification of their product topological space (hence their Cartesian product in the category of compactly generated topological spaces) is again a CW-complex with attaching maps .
and so then the product topological space itself has CW-complex structure.
A CW-complex is a locally contractible topological space.
For instance (Hatcher, prop. A.4).
Every CW-complex is a paracompact topological space.
See for instance (Hatcher K-theory, appendix of section 1.2). For a discussion that highlights which choice principles are involved, see (Fritsch-Piccinini 90, Theorem 1.3.5 (p. 29 and following)).
Every CW-complex is a compactly generated topological space.
Since a CW-complex is a colimit in Top over attachments of standard n-disks (its cells), by the characterization of colimits in (prop.) a subset of is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the -disks are compact, this implies one direction: if a subset of intersected with all compact subsets is closed, then is closed.
Every CW complex is homotopy equivalent to a space that admits a good open cover.
For instance (Hatcher, prop. A.5).
e.g. (AGP 02, def. 5.1.11)
Let be a CW-complex and write
for its filtered topological space-structure with the topological space obtained from by gluing on -cells. For write for the set of -cells of .
The relative singular homology of the filtering degrees is
where denotes the free abelian group on the set of -cells.
The proof is spelled out at Relative singular homology - Of CW complexes.
With we have
In particular if is a CW-complex of finite dimension (the maximum degree of cells), then
Moreover, for the inclusion
is an isomorphism and for we have an isomorphism
This is mostly for instance in (Hatcher, lemma 2.34 b),c)).
Now by prop. 6 the leftmost and rightmost homology groups here vanish when and and hence exactness implies that
Finally for the last claim use that the above exact sequence gives
and hence that with the above the map is surjective.
Basic textbook accounts include
Brayton Gray, Homotopy Theory: An Introduction to Algebraic Topology, Academic Press, New York (1975).
George Whitehead, chapter II of Elements of homotopy theory, 1978
Rudolf Fritsch and Renzo A. Piccinini, Cellular structures in topology, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (pdf)
Original articles include
John Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280.