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 no longer consistently interesting workout one has to head via sooner than you'll learn nice works of literature within the unique language.

The current e-book grew out of notes for an introductory topology path on the college 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 historical past in calculus and user-friendly algebra, yet now not inevitably in actual or advanced analysis.

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

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Extra resources for A Taste of Topology (Universitext)

Example text

A) For each x ∈ U , let Ix be the union of all open intervals contained in U and containing x. Show that Ix is an open (possibly unbounded) interval. (b) For x, y ∈ U , show that Ix = Iy or Ix ∩ Iy = ∅. (c) Conclude that U is a union of countably many, pairwise disjoint open intervals. 4. Let (X, d) be a metric space, and let S ⊂ X. The distance of x ∈ X to S is defined as dist(x, S) := inf{d(x, y) : y ∈ S} (where dist(x, S) = ∞ if S = ∅). Show that S = {x ∈ X : dist(x, S) = 0}. 5. Let Y be the subspace of B(N, F) consisting of those sequences tending to zero.

Let (X, d) be a metric space, and let Y be a subspace of X. Show that U ⊂ Y is open in Y if and only if there is V ⊂ X that is open in X such that U = Y ∩ V . 3 Convergence and Continuity The notion of convergence in Rn carries over to metric spaces almost verbatim. 1. Let (X, d) be a metric space. A sequence (xn )∞ n=1 in X is said to converge to x ∈ X if, for each > 0, there is n ∈ N such that d(xn , x) < for all n ≥ n . We then say that x is the limit of (xn )∞ n=1 and write x = limn→∞ xn or xn → x.

This is accomplished by letting d˜0 := d0 and, once d˜0 , . . , d˜n−1 have been defined for some n ∈ N, letting d˜n (x, y) := dn (x, y) + d˜n−1 (fn (x), fn (y)) (x, y ∈ Xn ). In what follows, we consider the spaces X0 , X1 , X2 , . . equipped with the metrics d˜0 , d˜1 , d˜2 , . . instead of with d0 , d1 , d2 , . .. Let U0 ⊂ X be open and not empty. We need to show that ∞ U0 ∩ (f1 ◦ · · · ◦ fn )(Xn ) = ∅. n=1 Since f1 (X1 ) is dense in X0 , there is x1 ∈ X1 with f1 (x1 ) ∈ U0 . Since f1 is continuous at x1 , there is δ1 ∈ (0, 1] such that f1 (Bδ1 (x1 )) ⊂ U0 .

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