Download A Taste of Topology by Volker Runde (auth.), S Axler, K.A. Ribet (eds.) PDF

By Volker Runde (auth.), S Axler, K.A. Ribet (eds.)

If arithmetic is a language, then taking a topology path on the undergraduate point is cramming vocabulary and memorizing abnormal verbs: an important, yet now not consistently intriguing workout one has to head via sooner than you can still learn nice works of literature within the unique language.

The current booklet grew out of notes for an introductory topology direction on the collage of Alberta. It offers a concise creation to set-theoretic topology (and to a tiny bit of algebraic topology). it really is available to undergraduates from the second one 12 months on, yet even starting graduate scholars can make the most 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 historical past in calculus and straightforward algebra, yet now not inevitably in actual or complicated analysis.

In a few issues, the booklet treats its fabric otherwise than different texts at the subject:

* Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem;

* Nets are used broadly, particularly for an intuitive facts of Tychonoff's theorem;

* a quick and stylish, yet little identified evidence for the Stone-Weierstrass theorem is given.

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Additional resources for A Taste of Topology

Example text

How does · ∞ relate to the metric D from the previous exercise? 4. Let (E, · ) be a normed space, and define ||| · ||| : E → [0, ∞) by letting |||x||| := x 1+ x (x ∈ E). Is ||| · ||| a norm on E? 5. 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/≈.

5. 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}.

Let (X, d) be a metric space. A sequence (xn )∞ n=1 in X is called a Cauchy sequence if, for each > 0, there is n > 0 such that d(xn , xm ) < for all n, m ≥ n . As in Rn , we have the following. 2. Let (X, d) be a metric space, and let (xn )∞ n=1 be a convergent sequence in X. Then (xn )∞ n=1 is a Cauchy sequence. Proof. Let x := limn→∞ xn , and let > 0. Then there is n > 0 such that d(xn , x) < 2 for all n ≥ n . Consequently, we have d(xn , xm ) ≤ d(xn , x) + d(x, xm ) < 2 + 2 (n, m ≥ n ), = so that (xn )∞ n=1 is a Cauchy sequence.

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