By John B. Conway

ISBN-10: 3319023675

ISBN-13: 9783319023670

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in element set topology is aimed toward an upper-undergraduate viewers. Its mild speed may be necessary to scholars who're nonetheless studying to put in writing proofs. must haves comprise calculus and no less than one semester of research, the place the scholar has been competently uncovered to the guidelines of easy set conception similar to subsets, unions, intersections, and services, in addition to convergence and different topological notions within the genuine line. Appendices are incorporated to bridge the space among this new fabric and fabric present in an research direction. Metric areas are one of many extra everyday topological areas utilized in different components and are for this reason brought within the first bankruptcy and emphasised through the textual content. This additionally conforms to the method of the ebook to begin with the actual and paintings towards the extra normal. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of thought, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact spaces.

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In fact, this example is a special case of the preceding one if we let d be the discrete metric on X. (c) If X is any set and T = {∅, X}, then T is a topology on X called the trivial topology. This topology does not arise from a metric if X has at least two points. B. 1007/978-3-319-02368-7 2, © Springer International Publishing Switzerland 2014 39 40 2. Topological Spaces (d) Let {(Xi , Ti ) : i ∈ I} be a collection of topological spaces, where the sets {Xi : i ∈ I} are pairwise disjoint subsets of some larger set that we can take to be X = i∈I Xi .

Xn in E such that: (i) for 1 ≤ k ≤ n, B(xk ; ) ⊆ E; (ii) for 2 ≤ k ≤ n, xk−1 ∈ B(xk ; ); (iii) x1 = x and xn = y. The concept of an -chain has limited value in an arbitrary metric space. For one thing, note that conditions (i) and (ii) of the definition imply that only points in the interior of E can be linked by an -chain. 14. If r > 0 and z ∈ Rq , then for any pair of points x and y in B(z; r) and all sufficiently small there is an -chain in B(z; r) from x to y. In fact, observe that the straight line segment from x to y is contained in B(z; r), and using this it is easy to construct the -chain.

Note that the metric used for Rq in the Heine–Borel Theorem must be the standard one, or the result may fail even for an equivalent metric. For example, if we use the metric d(x, y) = |x − y|(1 + |x − y|)−1 on R, then all of R is closed and bounded but not compact. This furnishes another example of a closed and bounded set that is not compact. 10. If (X, d) is a compact metric space and f : X → (Z, ρ) is a continuous function, then f is uniformly continuous. Proof. Suppose f is not uniformly continuous.

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A Course in Point Set Topology by John B. Conway

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