By Ronald Brown

The booklet is meant as a textual content for a two-semester direction in topology and algebraic topology on the complex undergraduate or starting graduate point. There are over 500 workouts, 114 figures, a number of diagrams. the overall course of the ebook is towards homotopy concept with a geometrical perspective. This booklet would supply a greater than enough heritage for the standard algebraic topology direction that starts with homology conception. for additional information see www.bangor.ac.uk/r.brown/topgpds.html This model dated April 19, 2006, has a few corrections made.

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4. Let f be the function x −→ x x−1 0 x 2

Suppose F : X × X → X is a map, and consider the function G:X×X→X (x, y) → F(y, F(x, y)). Then G is a map since it is the composite of the maps T ×1 1×∆ 1×F F X × X −→ X × X × X −→ X × X × X −→ X × X −→ X (x, y) → (x, y, y) → (y, x, y) → (y, F(x, y)) → G(x, y). 3. Let S1 = {(x, y) ∈ R2 : x2 + y2 = 1}, and consider the function f : [0, 1[ → S1 t → (cos 2πt, sin 2πt). Then f is a bijection, and it is continuous since its components are continuous. However f−1 is not continuous since M = [0, 21 ] is a neighbourhood in [0, 1[ of 0, but f[M] is not a neighbourhood in S1 of f(0) = (1, 0).

What is the P-adic topology in R if (a) P = R, (b) P = {0}? 8. Prove that Axioms N1–N4 are independent. 2 Open sets Let N be a neighbourhood topology on the set X. A subset U of X is open (with respect to N) if U is a neighbourhood of each of its points. Thus U is open if and only if U = Int U. 1 Let x ∈ X and N ⊆ X; N is a neighbourhood of x if and only if there is an open set U such that x ∈ U ⊆ N. Proof If N is a neighbourhood of x, then Int N is an open set such that x ∈ Int N ⊆ N. Conversely, if U is an open set such that x ∈ U ⊆ N, then U is a neighbourhood of x and hence so also is N.