By Ronald Brown
The publication is meant as a textual content for a two-semester direction in topology and algebraic topology on the complicated undergraduate or starting graduate point. There are over 500 routines, 114 figures, a variety of diagrams. the overall path of the e-book is towards homotopy idea with a geometrical viewpoint. This booklet would offer a greater than sufficient history for the standard algebraic topology direction that starts off with homology conception. for additional information see www.bangor.ac.uk/r.brown/topgpds.html This model dated April 19, 2006, has a couple of corrections made.
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Example text
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.