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

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

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 continuously interesting workout one has to head via earlier than you may learn nice works of literature within the unique language.

The current booklet grew out of notes for an introductory topology path 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 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 kids who've a historical past in calculus and effortless algebra, yet now not unavoidably in actual or complicated analysis.

In a few issues, the e-book treats its fabric otherwise than different texts at the subject:

* Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem;

* Nets are used generally, specifically for an intuitive evidence of Tychonoff's theorem;

* a brief and stylish, yet little identified evidence for the Stone-Weierstrass theorem is given.

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It is probably not exaggerated to say that most of functional analysis and abstract algebra would collapse without the axiom of choice. As Cantor proved, ℵ0 < c holds, and he himself already asked if there was any cardinal strictly between ℵ0 and c. The belief that no such cardinal exists is called the continuum hypothesis. His failure to prove it troubled Cantor deeply. In 1900, David Hilbert gave a famous speech at the International Congress of Mathematicians in Paris, in which he identified twenty-three open problems as central to mathematical research in the coming century; among them was the question of whether the continuum hypothesis was true.

Let U be an open cover of X. Then, for each j = 1, . . , n, there is Uj ∈ U such that xj ∈ Uj . It follows that S ⊂ U1 ∪ · · · ∪ Un . Hence, S is compact. (b) Let (X, d) be a compact metric space, and let ∅ = K ⊂ X be compact. Fix x0 ∈ K. Since {Br (x0 ) : r > 0} is an open cover of K, there are r1 , . . , rn > 0 such that K ⊂ Br1 (x0 ) ∪ · · · ∪ Brn (x0 ). With R := max{r1 , . . , rn }, we see that K ⊂ BR (x0 ), so that diam(K) ≤ 2R < ∞. This means, for example, that any unbounded subset of Rn (or, more generally, of any normed space) cannot be compact.

A) A subset D of X is said to be dense in X if D = X. (b) If X has a dense countable subset, then X is called separable. 16. (a) Q is dense in R. In particular, R is separable. 32 2 Metric Spaces (b) A subset S of a discrete metric space (X, d) is dense if and only if S = X. In particular, X is separable if and only if it is countable. The following hereditary property of separability is somewhat surprising, but very useful. 17. Let (X, d) be a separable metric space, and let Y be a subspace of X.

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