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By Volker Runde

If arithmetic is a language, then taking a topology direction on the undergraduate point is cramming vocabulary and memorizing abnormal verbs: an important, yet no longer constantly fascinating workout one has to head via sooner than you'll be able to learn nice works of literature within the unique language.

The current publication grew out of notes for an introductory topology path on the collage of Alberta. It presents a concise advent to set-theoretic topology (and to a tiny bit of algebraic topology). it truly is obtainable to undergraduates from the second one 12 months on, yet even starting graduate scholars can take advantage of a few parts.

Great care has been dedicated to the choice of examples that aren't self-serving, yet already available for college students who've a history in calculus and effortless algebra, yet now not inevitably in genuine or advanced analysis.

In a few issues, the booklet treats its fabric in a different way than different texts at the subject:
* Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem;
* Nets are used widely, specifically for an intuitive evidence of Tychonoff's theorem;
* a quick and chic, yet little recognized facts for the Stone-Weierstrass theorem is given.

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Hence, if every continuous function on [0, 2] ∞ is differentiable at some point of [0, 1], we have C([0, 2], R) = n=1 Fn . 17, we show that this is not possible. 17, we first need to show that the sets Fn for n ∈ N are closed in C([0, 2], R). Fix n ∈ N, and let (fm )∞ m=1 be a sequence in Fn such that fm − f ∞ → 0 for some f ∈ C([0, 2], R). For each m ∈ N, there is tm ∈ [0, 1] such that |fm (tm + h) − fm (tm )| ≤ n. h h∈(0,1) sup Suppose without loss of generality that (tm )∞ m=1 converges to some t ∈ [0, 1] (otherwise, replace (tm )∞ m=1 by a convergent subsequence).

4 Completeness 43 D(fn , f ) < 3 for n ≥ n . Fix n ≥ n . Since fn is continuous at x0 , the set N := fn−1 (B 3 (fn (x0 ))) is a neighborhood of x0 . Let x ∈ N , and note that d(f (x), f (x0 )) ≤ d(f (x), fn (x)) + d(fn (x), fn (x0 )) + d(fn (x0 ), f (x0 )) ≤ D(fn , f ) + d(fn (x), fn (x0 )) + D(fn , f ) 2 + d(fn (x), fn (x0 )), because n ≥ n , < 3 < , because x ∈ N. It follows that N ⊂ f −1 (B (f (x0 ))), so that f −1 (B (f (x0 ))) ∈ Nx0 . Since > 0 was arbitrary, this is enough to guarantee the continuity of f at x0 .

Let X be any set, and let d : X × X → [0, ∞) be a semimetric. For x, y ∈ X, define x ≈ y if and only if d(x, y) = 0. (a) Show that ≈ is an equivalence relation on X. (b) For x ∈ X, let [x] denote its equivalence class with respect to ≈, and let X/≈ denote the collection of all [x] with x ∈ X. Show that (X/≈) × (X/≈) → [0, ∞), defines a metric on X/≈. 1. Let (X, d) be a metric space, let x0 ∈ X, and let r > 0. The open ball centered at x0 with radius r is defined as Br (x0 ) := {x ∈ X : d(x, x0 ) < r}.

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