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 continually intriguing workout one has to head via sooner than you'll be able to learn nice works of literature within the unique language.
The current publication grew out of notes for an introductory topology path on the college of Alberta. It offers a concise advent to set-theoretic topology (and to a tiny bit of algebraic topology). it's available to undergraduates from the second one 12 months 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 students who've a historical past in calculus and basic algebra, yet now not unavoidably in actual or complicated analysis.
In a few issues, the ebook 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, specifically for an intuitive facts of Tychonoff's theorem;
* a quick and chic, yet little identified facts for the Stone-Weierstrass theorem is given.
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Extra resources for A Taste of Topology
Let (X, dX ) and (Y, dY ) be metric spaces such that (X, dX ) is discrete, and let f : X → Y be arbitrary. Let U ⊂ Y be open. Since in a discrete space every set is open, it follows that f −1 (U ) is open. Consequently, f must be continuous. As we have seen, there can be diﬀerent metrics on one set. For many purposes, it is convenient to view certain metrics as identical. 12. Let X be a set. Two metrics d1 and d2 on X are said to be equivalent if the identity map on X is continuous both from (X, d1 ) to (X, d2 ) and from (X, d2 ) to (X, d1 ).
Hence, there are U1 , . . , Un ∈ U with K = f −1 (U1 ) ∪ · · · ∪ f −1 (Un ) and thus f (K) ⊂ U1 ∪ · · · ∪ Un . This proves the claim. 6. Let (K, d) be a non-empty, compact metric space, and let f : K → R be continuous. Then f attains both a minimum and a maximum on K. Proof. Let M := sup f (K). Since f (K) is compact, it is bounded, so that M < ∞. For each n ∈ N, there is yn ∈ f (K) such that yn > M − n1 ; it is clear that M = limn→∞ yn . Since f (K) is closed in R, it follows that M ∈ f (K). Hence, there is x0 ∈ K such that f (x0 ) = M .
This makes the following deﬁnition signiﬁcant. 3. A metric space (X, d) is called complete if every Cauchy sequence in X converges. A normed space that is complete with respect to the metric induced by its norm is also called a Banach space. 4. (a) Rn is complete. (b) In a discrete metric space, every Cauchy sequence is eventually constant and therefore convergent. Hence, discrete metric spaces are complete. (c) Let S = ∅ be a set, and let (Y, d) be a complete metric space. 2(d) is complete. Let > 0, and choose n > 0 (fn )∞ n=1 be a Cauchy sequence in B(S, Y ).