Download A Taste of Topology (Universitext) by Volker Runde PDF

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 constantly fascinating workout one has to head via ahead of you may 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 presents a concise creation to set-theoretic topology (and to a tiny bit of algebraic topology). it's obtainable 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 obtainable for college students who've a heritage in calculus and simple algebra, yet no longer unavoidably in actual or complicated analysis.

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

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2. Let (X, d) be a complete metric space, and let (xn )∞ n=0 be a sequence in X such that there is θ ∈ (0, 1) with d(xn+1 , xn ) ≤ θ d(xn , xn−1 ) for n ∈ N. Show that (xn )∞ n=0 is convergent. 3. Use the previous problem to prove Banach’s fixed point theorem: if (X, d) is a complete metric space, and if f : X → X is such that d(f (x), f (y)) ≤ θ d(x, y) (x, y ∈ X) for some θ ∈ (0, 1), then there is a unique x ∈ X with f (x) = x. 4. Let (X, d) be a metric space, and let ∅ = S ⊂ X. Show that diam(S) = inf{r > 0 : S ⊂ Br (x) for all x ∈ S}.

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. 3. A Hamel basis of a (possibly infinite-dimensional) vector space (over an arbitrary field) is a linearly independent subset whose linear span is the whole space.

14. 15. Let (X, d) be a metric space, and let U1 , . . , Un ⊂ X be dense open subsets of X. Then U1 ∩ · · · ∩ Un is dense in X. Proof. By induction, it is clear that we may limit ourselves to the case where n = 2. Let x ∈ X, and let > 0. Since U1 is dense in X, we have B (x) ∩ U1 = ∅. Since B (x)∩U1 is open—and thus a neighborhood of each of its points—it follows from the denseness of U2 that B (x) ∩ U1 ∩ U2 = ∅. Since > 0 was arbitrary, we conclude that x ∈ U1 ∩ U2 . 16 (Baire’s theorem). Let (X, d) be a complete metric space, and let (Un )∞ n=1 be a sequence of dense open subsets of X.

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