By John B. Conway

This textbook in aspect set topology is geared toward an upper-undergraduate viewers. Its light velocity may be precious to scholars who're nonetheless studying to write down proofs. necessities comprise calculus and a minimum of one semester of research, the place the coed has been correctly uncovered to the information of simple set conception comparable to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the actual line. Appendices are incorporated to bridge the distance among this new fabric and fabric present in an research direction. Metric areas are one of many extra commonly used topological areas utilized in different components and are consequently brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the procedure of the ebook to begin with the actual and paintings towards the extra common. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of suggestion, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued features, culminating in a improvement of paracompact areas.

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**Example text**

15(b). Thus, the assumption there that G is open is essential. We close this section with a circumspective remark. We did not use sequences in this section. Connectedness is one of the only properties of a metric space I know whose examination never uses the concept of a convergent sequence. 2. 2, that a subset E of R is an interval if and only if when a, b ∈ E and a < b, then c ∈ E whenever a < c < b? (3) If (X, d) is connected and f : X → R is a continuous function such that |f (x)| = 1 for all x in X, show that f must be constant.

9 (Urysohn’s4 Lemma). If A and B are two disjoint closed subsets of X, then there is a continuous function f : X → R having the following properties: (a) 0 ≤ f (x) ≤ 1 for all x in X: (b) f (x) = 0 for all x in A: (c) f (x) = 1 for all x in B. Proof. Define f : X → R by f (x) = dist (x, A) , dist (x, A) + dist (x, B) which is well defined since the denominator never vanishes. ) It is easy to check that f has the desired properties. 10. If F is a closed subset of X and G is an open set containing F , then there is a continuous function f : X → R such that 0 ≤ f (x) ≤ 1 for all x in X, f (x) = 1 when x ∈ F , and f (x) = 0 when x ∈ / G.

While he was there, World War I began, and he was unable to return to France. His health deteriorated further, depression ensued, and he spent the rest of his life on the shores of Lac L´ eman in Switzerland. It was there that he received the Chevalier de la Legion d’Honneur, and in 1922 he was elected to the Acad´ emie des Sciences. He published significant works on number theory and functions. He died in 1932 at Chamb´ ery near Geneva. Exercises 37 The details of this induction argument are left as Exercise 1.