By Rudolf Fritsch

This e-book describes the development and the homes of CW-complexes. those areas are vital simply because to start with they're the right kind framework for homotopy thought, and secondly so much areas that come up in natural arithmetic are of this kind. The authors speak about the principles and likewise advancements, for instance, the speculation of finite CW-complexes, CW-complexes when it comes to the speculation of fibrations, and Milnor's paintings on areas of the kind of CW-complexes. They identify very truly the connection among CW-complexes and the idea of simplicial complexes, that's constructed in nice element. routines are supplied during the e-book; a few are hassle-free, others expand the textual content in a non-trivial manner. For the latter; additional reference is given for his or her resolution. each one bankruptcy ends with a bit sketching the old improvement. An appendix provides uncomplicated effects from topology, homology and homotopy idea. those positive factors will reduction graduate scholars, who can use the paintings as a direction textual content. As a latest reference paintings it is going to be crucial analyzing for the extra really good staff in algebraic topology and homotopy thought.

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Unfortunately, as experiment will show, the whole thing gets hopelessly tangled. The point is, that this sort of model making is impossible in R3 —an extra dimension is needed. ) S2 \ D D1 b b D1 E M E Fig. 8 The question is: can we anyway say what we mean by this stitching process without having to produce the result as a subset of R3 ? One of the properties of the model we should like is that if in Fig. 4] 17 S2 is identified with b in M, then the curve shown should be continuous. This can be arranged by defining neighbourhoods suitably.

For each λ ∈ U there is a basic neighbourhood M × N of λ such that M × N ⊆ U. Let Uλ = Int M, Vλ = Int N. Then Uλ , Vλ are open and U = λ∈U Uλ × Vλ . ✷ E XAMPLE Let α = (a, b) ∈ R2 , and let r > 0. The open ball about α of radius r is the set B(α, r) = {(x, y) ∈ R2 : (x − a)2 + (y − b)2 < r2 }. This open ball is an open set: For, let α = (a , b ) ∈ B(α, √r) and let s = (a − a)2 + (b − b)2 . Then s < r. Let 0 < δ < (r − s)/ 2, M = ]a − δ, a + δ[, N = ]b − δ, b + δ[. Then M × N ⊆ B(α, r) and so B(α, r) is a neighbourhood of α .

6 Let f : Z → X, g : Z → Y be maps. Then (f, g) : Z → X × Y is a map. Proof Let h = (f, g), so that h sends z → (f(z), g(z)). Let P be a neighbourhood of h(z), and let M × N be a basic neighbourhood of h(z) contained in P. Then h−1 [P] contains the set h−1 [M × N] = {z ∈ Z : f(z) ∈ M, g(z) ∈ N} = f−1 [M] ∩ g−1 [N]. It follows that h−1 [P] is a neighbourhood of z. This result can also be expressed: a function h : Z → X × Y is continuous ⇔ p1 h, p2 h are continuous. 6 since p1 h, p2 h are the components f, g of h.