By Ronald Brown

The publication 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 routines, 114 figures, a number of diagrams. the overall course of the publication is towards homotopy idea with a geometrical perspective. This booklet would supply a greater than enough heritage for the standard algebraic topology direction that starts with homology concept. 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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Let l ∈ R and let f : R → R be the constant function x → l. Let a ∈ R. The domain of f is R, which is a neighbourhood of a. If N is a neighbourhood of l, then f−1 [N] = R, which is a neighbourhood of a. Therefore, f is continuous at a, and since a is arbitrary, f is a continuous function. 2. Let f : R → R be the identity function x → x. Let a ∈ R. The domain of f is a neighbourhood of a, and f(a) = a. If N is a neighbourhood of f(a), then f−1 [N] = N, so that f−1 [N] is a neighbourhood of a. Thus f is a continuous function.

A) N is a neighbourhood of x ∈ X if x ∈ N ⊆ X and X \ N is finite. (b) N is a neighbourhood of x ∈ X if x ∈ N ⊆ X and X \ N is countable. Can either of these topologies be the discrete topology? 7. Let X = Z and let p be a fixed integer. A set N ⊆ Z is a p-adic neighbourhood of n ∈ Z if N contains the integers n + mpr for some r and all m = 0, ±1, ±2, . . (so that in a given neighbourhood r is fixed but m varies). Prove that the p-adic neighbourhoods form a neighbourhood topology on Z, the p-adic topology.

1 f is continuous ⇔ for each x in X and N ∈ B (f(x)), there is an M ∈ B(x) such that f[M] ⊆ N. 6] 39 Proof The proof is simple. ⇒ Let x ∈ X, N ∈ B (f(x)). Then f−1 [N] is a neighbourhood of x and so there is an M ∈ B(x) such that M ⊆ f−1 [N]. This implies f[M] ⊆ N. ⇐ Let x ∈ X and let P be a neighbourhood of f(x). Then there exists N ∈ B (f(x)) such that N ⊆ P. By assumption there is an M ∈ B(x) such that M ⊆ f−1 [N]. So f−1 [P], which contains f−1 [N], is a neighbourhood of x. 6. The continuity of a function can also be described in terms of open sets, closed sets, or closure.