By John B. Conway

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its light speed should be invaluable to scholars who're nonetheless studying to write down proofs. must haves comprise calculus and at the very least one semester of study, the place the coed has been thoroughly uncovered to the information of uncomplicated set concept reminiscent of subsets, unions, intersections, and features, 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 path. Metric areas are one of many extra accepted topological areas utilized in different components and are for that reason brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the procedure of the booklet 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 proposal, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact areas.

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**Additional info for A Course in Point Set Topology (Undergraduate Texts in Mathematics)**

**Example text**

8) If (X1 , d1 ), (X2 , d2 ) are metric spaces, show that X1 × X2 is compact if and only if both X1 and X2 are compact. (9) Give an example of a noncompact metric space (X, d) and a continuous function f : (X, d) → (Z, ρ) such that f (X) is compact. (10) For two subsets A and B of X, define the distance from A to B by dist (A, B) = inf{d(a, b) : a ∈ A, b ∈ B}. (a) Show that dist (A, B) = dist (B, A) = dist (cl A, cl B). (b) If A and B are two disjoint closed subsets of X such that B is compact, then dist (A, B) > 0.

7(a), but we must manufacture an actual subsequence of the original sequence. This takes a little bit of care and eﬀort, which we leave to the interested reader. (c) implies (d). If S is an infinite subset, then S has a sequence of distinct points {xn }; by (c), there is a subsequence {xnk } that converges to some point x. It follows that x is a limit point of S. ) (a) implies (d). Assume that (d) is false. So there is an infinite subset S of K with no limit point; it follows that there is an infinite sequence {xn } in S with no limit point.

He died in 1956 in Paris. 5. Connectedness 29 Proof. For each natural number n we can find a finite set Fn such that X = {B(x; n−1 ) : x ∈ Fn }. Put F = ∞ n=1 Fn ; we will show that this countable set F is dense in X. In fact, if x0 is an arbitrary point in X and ϵ > 0, then choose n such that n−1 < ϵ. Thus, there is a point x in Fn ⊆ F with d(x0 , x) < n−1 < ϵ, proving that x0 ∈ cl F . Exercises (1) Show that the union of a finite number of compact sets is compact. (2) If K is a subset of (X, d), show that K is compact if and only if every cover of K by relatively open subsets of K has a finite subcover.

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

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