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

ISBN-10: 038725790X

ISBN-13: 9780387257907

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 regularly interesting workout one has to move via sooner than possible learn nice works of literature within the unique language.

The current publication grew out of notes for an introductory topology direction on the collage of Alberta. It presents a concise advent to set-theoretic topology (and to a tiny bit of algebraic topology). it truly is 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 available for college students who've a history in calculus and ordinary algebra, yet no longer unavoidably in genuine or complicated 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 widely, specifically for an intuitive evidence of Tychonoff's theorem;

* a brief and chic, yet little recognized facts for the Stone-Weierstrass theorem is given.

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**Example text**

Let x ∈ N , and note that d(f (x), f (x0 )) ≤ d(f (x), fn (x)) + d(fn (x), fn (x0 )) + d(fn (x0 ), f (x0 )) ≤ D(fn , f ) + d(fn (x), fn (x0 )) + D(fn , f ) 2 + d(fn (x), fn (x0 )), because n ≥ n , < 3 < , because x ∈ N. It follows that N ⊂ f −1 (B (f (x0 ))), so that f −1 (B (f (x0 ))) ∈ Nx0 . Since > 0 was arbitrary, this is enough to guarantee the continuity of f at x0 . 5, the following assertion seems to defy reason at ﬁrst glance. 7. Let (X, d) be a complete metric space, and let U ⊂ X be open.

Then there are several norms on E, for example, · 1 deﬁned by 1 f 1 := |f (t)| dt (f ∈ E) 0 or · ∞ given by f ∞ := sup{|f (t)| : t ∈ [0, 1]} (f ∈ E). Each of them turns E into a normed space. (d) Let S = ∅ be a set, and let (Y, d) be a metric space. A function f : S → Y is said to be bounded if sup d(f (x), f (y)) < ∞ x,y∈S The set B(S, Y ) := {f : S → Y : f is bounded} becomes a metric space through D deﬁned by D(f, g) := sup d(f (x), g(x)) x∈S (f, g ∈ B(S, Y )). 1 Deﬁnitions and Examples 25 (e) France is a centralized country: every train that goes from one French city to another has to pass through Paris.

1. 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.

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

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