A Course in Point Set Topology (Undergraduate Texts in by John B. Conway

By John B. Conway

This textbook in element set topology is aimed toward an upper-undergraduate viewers. Its light velocity can be precious to scholars who're nonetheless studying to write down proofs. necessities comprise calculus and no less than one semester of study, the place the scholar has been appropriately uncovered to the information of easy set concept equivalent 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 path. Metric areas are one of many extra known topological areas utilized in different parts and are accordingly brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the method of the publication 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 notion, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued services, culminating in a improvement of paracompact areas.

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Example text

Now we introduce more general cartesian products. Definition. Let A be a nonempty collection of sets. An indexing function for A is a surjective function f from some set J , called the index set, to A. The collection A, together with the indexing function f , is called an indexed family of sets. Given α ∈ J , we shall denote the set f (α) by the symbol Aα . ” Sometimes we write merely {Aα }, if it is clear what the index set is. Note that although an indexing function is required to be surjective, it is not required to be injective.

Then the set Z+ of positive integers is defined by the equation Z+ = A. A∈A Note that the set R+ of positive real numbers is inductive, for it contains 1 and the statement x > 0 implies the statement x + 1 > 0. Therefore, Z+ ⊂ R+ , so the elements of Z+ are indeed positive, as the choice of terminology suggests. Indeed, one sees readily that 1 is the smallest element of Z+ , because the set of all real numbers x for which x ≥ 1 is inductive. The basic properties of Z+ , which follow readily from the definition, are the following: (1) Z+ is inductive.

9. (a) The formula (∗) h(1) = 1, h(2) = 2, h(n) = [h(n + 1)]2 − [h(n − 1)]2 for n ≥ 2 is not one to which the principle of recursive definition applies. Show that nevertheless there does exist a function h : Z+ → R satisfying this formula. ] (b) Show that the formula (∗) of part (a) does not determine h uniquely. ] (c) Show that there is no function h : Z+ → R satisfying the formula h(1) = 1, h(2) = 2, h(n) = [h(n + 1)]2 + [h(n − 1)]2 ∗ §8 for n ≥ 2. 2. That should make the underlying idea of the proof much clearer when we consider the general case.

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