By John B. Conway

ISBN-10: 3319023675

ISBN-13: 9783319023670

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in aspect set topology is aimed toward an upper-undergraduate viewers. Its light velocity can be invaluable to scholars who're nonetheless studying to put in writing proofs. necessities contain calculus and a minimum of one semester of study, the place the coed has been appropriately uncovered to the information of simple set concept similar to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the actual line. Appendices are integrated to bridge the distance among this new fabric and fabric present in an research direction. Metric areas are one of many extra typical topological areas utilized in different components and are consequently brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the strategy 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 services, culminating in a improvement of paracompact spaces.

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**Extra resources for A Course in Point Set Topology**

**Example text**

Condition (i) of the deﬁnition of an -chain implies B ⊆ G, and so B ⊆ H. In particular, y ∈ H. Since y was arbitrary, H = G and G is connected. Now assume that G is connected. Fix a point x in G, and let D = {y ∈ G : there is an > 0 and an -chain in G from x to y} The strategy of the proof will be to show that D is both relatively open and closed in G; since it is not empty (x ∈ D), it will then follow that D = G, and so G will have been shown to satisfy the condition. If y ∈ D, then let > 0 and let x1 , .

Proof. If f : X → Z and g : Z → W , then for any subset G of W , then we have that (g ◦ f )−1 (G) = f −1 [g −1 (G)]. Thus, if G is an open subset of W , it follows that (g ◦ f )−1 (G) is open in X. Hence g ◦ f is continuous. Later in the book we will present several results on manufacturing continuous functions from a metric space into the real numbers. We have seen examples of such continuous functions on speciﬁc metric spaces, but we want to show the existence of continuous functions with speciﬁc properties on arbitrary ones.

B) If X is any set and T = 2X , the collection of all subsets of X, then (X, T ) is a topological space. This topology is called the discrete topology on X. 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.

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