Read e-book online A Taste of Topology (Universitext) PDF

By Volker Runde

ISBN-10: 0387283870

ISBN-13: 9780387283876

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 interesting workout one has to head via sooner than you will 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 offers a concise creation to set-theoretic topology (and to a tiny bit of algebraic topology). it truly is available to undergraduates from the second one yr on, yet even starting graduate scholars can reap the benefits 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 straight forward algebra, yet no longer inevitably in actual or complicated analysis.

In a few issues, the publication 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, particularly for an intuitive evidence of Tychonoff's theorem;
* a brief and chic, yet little identified facts for the Stone-Weierstrass theorem is given.

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Example text

Then there is an open subset U of X contained in N with x ∈ U . , S ⊂ X \U ). Since X \ U is closed, it follows that S ⊂ X \ U and thus x ∈ X \ U , which is a contradiction. Consequently, x ∈ cl(S) holds. Conversely, let x ∈ cl(S), and assume that x ∈ / S. Then U := X \ S is an open set containing x (thus belonging to Nx ) having empty intersection with S. This contradicts x ∈ cl(S). 14. (a) Any open interval in R contains a rational number. Hence, we have Q = R. (b) Let (X, d) be any metric space.

Let S be any set, and let X consist of the finite subsets of S. Show that d : X × X → [0, ∞), (A, B) → |(A \ B) ∪ (B \ A)| is a metric on X. 2. 2(d) in detail. 3. Let S = ∅ be a set, and let E be a normed space. Show that f ∞ := sup{ f (x) : x ∈ S} (f ∈ B(S, E)) defines a norm on B(S, E). 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.

We claim that the map ι : X → Cb (X, R), x → fx is an isometry. To see this, fix x, y ∈ X and note that, by (∗ ∗ ∗) again, D(ι(x), ι(y)) = sup |fx (t) − fy (t)| = sup |d(x, t) − d(y, t)| ≤ d(x, y), t∈X t∈X holds; on the other hand, we have D(ι(x), ι(y)) = sup |fx (t) − fy (t)| ≥ |fx (y) − fy (y)| = d(x, y), t∈X which proves the claim. 4 Completeness 47 In view of the uniqueness of a completion up to isometric isomorphism, we are justified to speak of the completion of a metric space. For the sake of notational convenience, we also identify a metric space with its canonical image in its completion.

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A Taste of Topology (Universitext) by Volker Runde


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