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

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

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 constantly intriguing workout one has to head via prior to you can actually learn nice works of literature within the unique language.

The current ebook grew out of notes for an introductory topology path on the collage of Alberta. It presents a concise creation to set-theoretic topology (and to a tiny bit of algebraic topology). it really is available to undergraduates from the second one 12 months 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 historical past in calculus and effortless algebra, yet now not inevitably in genuine 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 largely, particularly for an intuitive facts of Tychonoff's theorem;

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

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Extra info for A Taste of Topology

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Then: (i) ∅ and X are closed; (ii) If F is a family of closed subsets of X, then {F : F ∈ F } is closed; (iii) If F1 and F2 are closed subsets of X, then F1 ∪ F2 is closed. Of course, in most metric spaces there are many sets that are neither open nor closed. Nevertheless, we can make the following definition. 12. Let (X, d) be a metric space. For each S ⊂ X, the closure of S is defined as S := {F : F ⊂ X is closed and contains S}. 11(ii) it is immediate that the closure of a set is a closed set.

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. A subset N of X is called a neighborhood of x if there is an open subset U of X with x ∈ U ⊂ N . The collection of all neighborhoods of x is denoted by Nx . 8. Let (X, d) be a metric space, and let x ∈ X. Then: (i) A subset N of X belongs to Nx if and only if there is B (x) ⊂ N ; (ii) If N ∈ Nx and M ⊃ N , then M ∈ Nx ; (iii) If N1 , N2 ∈ Nx , then N1 ∩ N2 ∈ Nx . > 0 such that Moreover, a subset U of X is open if and only if U ∈ Ny for each y ∈ U .

4. Let (X, d) be a metric space, and let ∅ = S ⊂ X. Show that diam(S) = inf{r > 0 : S ⊂ Br (x) for all x ∈ S}. 5. Give an example showing that the demand that limn→∞ diam(F Tn ) = 0 in Cantor’s intersection theorem cannot be dropped if we still want ∞ n=1 Fn = ∅ to hold. 6. Let E be a normed space with a countable Hamel basis. Show that E is a Banach space if and only if dim E < ∞. ) 7. Let (fk )∞ k=1 be a sequence in C([0, 1], F) that converges pointwise to a function f : [0, 1] → F. (a) For θ > 0 and n ∈ N, let Fn := {t ∈ [0, 1] : |fn (t) − fk (t)| ≤ θ for all k ≥ n}.

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