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

By Volker Runde

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 possible learn nice works of literature within the unique language.

The current booklet 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 obtainable to undergraduates from the second one yr 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 heritage in calculus and easy 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 largely, specifically 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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As we have seen, there can be different metrics on one set. For many purposes, it is convenient to view certain metrics as identical. 12. Let X be a set. Two metrics d1 and d2 on X are said to be equivalent if the identity map on X is continuous both from (X, d1 ) to (X, d2 ) and from (X, d2 ) to (X, d1 ). 10, two metrics d1 and d2 on a set X are equivalent if and only if they yield the same open sets (or, equivalently, the same closed sets). 13. (a) The Euclidean metric on Rn and the discrete metric are not equivalent.

There is i0 ∈ I \ Jmax ). Fix x0 ∈ Si0 , and define f˜ : Jmax ∪ {i0 } → j∈Jmax Sj ∪ Si0 by letting f˜(j) := It follows that fmax (j), j ∈ Jmax , x0 , j = i0 . Jmax ∪ {i0 }, f˜ ∈ P with (Jmax , fmax ) but (Jmax , fmax ) = Jmax ∪ {i0 }, f˜ , Jmax ∪ {i0 }, f˜ , which contradicts the maximality of (Jmax , fmax ). Hence, Jmax = I holds, so that fmax ∈ i Si . 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).

But, of course, we know that [0, 1] is not open. How is this possible? The answer is that openness (as well as all the notions that are derived from it) depends on the context of a given metric space. Thus, [0, 1] is open in [0, 1], but not open in R. 6. Let (X, d) be a discrete metric space, and let S ⊂ X. Then {x} = S= x∈S B1 (x) x∈S is open; that is, all subsets of X are open. A notion closely related to open sets is that of a neighborhood of a point. 7. Let (X, d) be a metric space, and let x ∈ X.

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