By Christoph Schweigert

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Extra info for Algebraic Topology [Lecture notes]

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A continuous map f : X → Y is called cellular, if f (X n ) ⊂ Y n for all n 0. The category of CW complexes together with cellular maps is rather flexible. Most of the classical constructions do not lead out of it: for example, CW complexes nicely behave with respect to collapsing subspaces to points. If X is a CW complex and A ⊂ X a subcomplex, the cell decomposition of X/A consisting the zero-cell A and the cells of X \ A is again a CW decomposition. Thus X/A is a CW complex in a canonical way.

5. The unit interval [0, 1] has a CW structure with two zero cells and one 1-cell. But for 1 instance the decomposition σ00 = {0}, σk0 = { k1 }, k > 0 and σk1 = ( k+1 , k1 ) does not give a CW structure on [0, 1]. Consider the following subset A ⊂ [0, 1] A := 1 2 1 1 + k k+1 |k ∈ N . 1 Then A ∩ σ ˉk1 is precisely the point 12 ( k1 + k+1 ). This is closed, but A is not closed in [0, 1], since it does not contain the limit point 0 of A. 8. • Historically, the notion of a simplicial complex plays an important role: a set K of simplices in Rn is called a simplicial complex or polyhedron, if the following conditions are satisfied: (a) If K contains a simplex, it contains all faces of this simplex.

Every finite CW complex is compact, since it is the union of finitely many compact subspaces Φσ (Dn ). 7. 1. The CW structures on a fixed topological space are not unique. for example, S2 with the CW structure from the cell-decomposition S2 \ {N } {N } has a single 0-cell consisting of the north pole and one 2-cell. Projections of a tetrahedron, cube, octahedron or even less regular bodies to the sphere provide other CW structures. 46 2. Consider the following spaces with cell decomposition: Figure 1 has two 0-cells and two 1-cells.