A Taste of Topology (Universitext) by Volker Runde

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 now not continuously fascinating workout one has to head via ahead of you may learn nice works of literature within the unique language.

The current publication grew out of notes for an introductory topology direction on the college of Alberta. It offers a concise creation to set-theoretic topology (and to a tiny bit of algebraic topology). it's available 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 kids who've a history in calculus and user-friendly algebra, yet now not inevitably in genuine or complicated analysis.

In a few issues, the e-book 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 commonly, specifically for an intuitive facts of Tychonoff's theorem;
* a brief and stylish, yet little identified facts for the Stone-Weierstrass theorem is given.

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3. Let (X, d) be a metric space, let (xn )∞ m=1 be a sequence in X, and let x, x ∈ X be such that (xn )∞ converges to both x and x . Then n=1 x and x are equal. Proof. Assume that x = x , so that := 12 d(x, x ) > 0. Since xn → x, there is n1 ∈ N such that d(xn , x) < for n ≥ n1 , and since xn → x , too, there is n2 ∈ N such that d(xn , x ) < for n ≥ n2 . Let n := max{n1 , n2 }, so that d(x, x ) ≤ d(x, xn ) + d(xn , x ) < + = d(x, x ), which is nonsense. 3 in a sketch. ε x’ xn 2 ε x xn 1 xn Fig.

The following is an immediate consequence of Baire’s theorem (just pass to complements). 17. Let (X, d) be a complete metric space, and let (Fn )∞ n=1 be ∞ a sequence of closed subsets of X such that n=1 Fn has a nonempty interior. Then at least one of the sets F1 , F2 , . . has a nonempty interior. 4 Completeness 49 To illustrate the power of Baire’s theorem, we turn to an example from elementary calculus. We all know that there are continuous functions that are not differentiable at certain points (take the absolute value function, for instance), and it is not very hard to come up with continuous functions that are not differentiable at a finite, and even countable, number of points.

7. One does. 7—which is then called the axiom of choice—is true and then deduce Zorn’s lemma from it. Exercises 1. Let S = ∅ be a set. , if x, y, z ∈ S are such that (x, y), (y, z) ∈ R, then (x, z) ∈ R holds). ) Given x ∈ S, the equivalence class of x (with respect to a given equivalence relation R) is defined to consist of those y ∈ S for which (x, y) ∈ R. Show that two equivalence classes are either disjoint or identical. be a sequence of nonempty sets. Show without invoking Zorn’s 2. Let (Sn )∞ n=1 Q lemma that ∞ n=1 Sn is not empty.

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