What is a finite CW complex?

What is a finite CW complex?

A finite CW-complex is a CW-complex which admits a presentation in which there are a finite number of attaching maps. The homotopy type of a finite CW-complex is called a finite homotopy type.

What does CW mean in CW complex?

closure-finite
The C stands for “closure-finite”, and the W for “weak” topology. A CW complex can be defined inductively. A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).

Are manifolds CW complexes?

Every compact smooth manifold admits a smooth triangulation and hence a CW-complex structure.

Is every simplicial complex a CW complex?

Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex. Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called m-cofibrant spaces, is a convenient category of spaces for algebraic topology.

What is complex topology?

In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems.

Is a simplicial complex a CW complex?

The geometric realization of any simplicial set is a CW-complex (Milnor 57). In particular, in the context of the homotopy hypothesis the Quillen equivalence between ∞-groupoids and nice topological spaces maps each ∞-groupoid to a CW-complex.

Which is the best definition of a CW complex?

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. Given a CW-complex, then Xn is also called its n – skeleton.

Is the product of X and K a CW complex?

If X is a CW-complex and K is a finite CW-complex, then the product topological space X × K is naturally itself a CW-complex. For example the suspension of a CW-complex itself carries the structure of a CW-complex.

How is an infinite-dimensional CW complex constructed?

An infinite-dimensional CW complex can be constructed by repeating the above process countably many times. , a k-cell is the interior of a k -dimensional ball added at the k -th step. The k-skeleton of the complex is the union of all its k -cells. As mentioned above, every collection of discrete points is a CW complex (of dimension 0).

Which is homotopy equivalent to a CW complex?

Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called m-cofibrant spaces, is a convenient category of spaces for algebraic topology. Also, CW complexes are among the cofibrant objects in the classical model structure on topological spaces.