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 fascinating workout one has to move via earlier than you possibly can 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 creation to set-theoretic topology (and to a tiny bit of algebraic topology). it truly is available to undergraduates from the second one yr 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 kids who've a heritage in calculus and easy algebra, yet now not inevitably in actual or advanced analysis.

In a few issues, the ebook 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 largely, specifically for an intuitive evidence of Tychonoff's theorem;

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

**Read or Download A Taste of Topology (Universitext) PDF**

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**Sample text**

44 2 Metric Spaces This is clearly wrong. The apparent paradox is resolved if one recalls the deﬁnition of a subspace of a metric space: (U, dU ) is not a subspace of the metric space (X, d), even though the two metrics d and dU are equivalent on U . We now present a famous property of complete metric spaces, for which we ﬁrst require a deﬁnition. 8. Let (X, d) be a metric space. The diameter of a subset S = ∅ of X is deﬁned as diam(S) := sup{d(x, y) : x, y ∈ S}. 9 (Cantor’s intersection theorem).

Let (E, · ) be a normed space, let x0 ∈ E, and let r > 0. Then, intuitively, one might view the boundary of the open ball Br (x0 ) as the sphere Sr [x0 ] := {x ∈ E : x − x0 = r}. Let x ∈ Sr [x0 ], and let > 0. Let δ ∈ (0, 1) be such that δ x − x0 < , and let y := x0 + (1 − δ)(x − x0 ). 14(c), it follows that y ∈ B (x) ∩ Br (x0 ), so that B (x) ∩ Br (x0 ) = ∅ and B (x) ∩ (E \ Br (x0 )) = ∅. (∗∗) On the other hand, since Br (x0 ) and E \ Br [x0 ] are open, it follows that any element x of E satisfying (∗∗) for each > 0 must lie in Sr [x0 ].

We all know that there are continuous functions that are not diﬀerentiable at certain points (take the absolute value function, for instance), and it is not very hard to come up with continuous functions that are not diﬀerentiable at a ﬁnite, and even countable, number of points. But is there a continuous function, on an interval say, that fails to be diﬀerentiable at each point of its domain? The following example gives the answer. 18. For n ∈ N, let Fn consist of those f ∈ C([0, 2], R) for which there is t ∈ [0, 1] such that |f (t + h) − f (t)| ≤ n.