A Course in Point Set Topology by John B. Conway

By John B. Conway

This textbook in aspect set topology is aimed toward an upper-undergraduate viewers. Its light velocity may be worthwhile to scholars who're nonetheless studying to put in writing proofs. necessities contain calculus and not less than one semester of study, the place the coed has been thoroughly uncovered to the tips of uncomplicated set thought akin to subsets, unions, intersections, and features, in addition to convergence and different topological notions within the genuine line. Appendices are integrated to bridge the distance among this new fabric and fabric present in an research path. Metric areas are one of many extra ordinary topological areas utilized in different components and are as a result brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the procedure of the booklet first of all the actual and paintings towards the extra basic. 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 services, culminating in a improvement of paracompact spaces.

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Now let x1 , . . , xn ∈ K such that K ⊆ k=1 B(xk ; r), and for 1 ≤ k ≤ n let Gk ∈ G such that B(xk ; r) ⊆ Gk . {G1 , . . , Gn } is the sought-after finite subcover. (c) implies (e). 11, the original sequence converges. Thus (K, d) is complete. Now fix an r > 0. Let x1 ∈ K; if K ⊆ B(x1 ; r), then we are done. If not, then there is a point x2 in K\B(x1 ; r). Once again, if K ⊆ B(x1 ; r) ∪ B(x2 ; r), we are done; otherwise, pick an x3 in K\[B(x1 ; r) ∪ B(x2 ; r)]. Continue. If this process does not stop after a finite number of steps, then we produce an infinite sequence {xn } in K with d(xn , xm ) ≥ r whenever n = m.

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.

That is, via f they have the same convergent sequences, open sets, closed sets, and continuous functions. 1) between metric spaces. To be sure, it is not the only possible equivalence relation, and perhaps the idea of an isometry, a map f : (X, d) → (Z, ρ) satisfying ρ(f (x), f (y)) = d(x, y) for all x, y in X, seems more natural. Indeed, isometries identify the two structures, and it readily follows that an isometry is a homeomorphism. Nevertheless, we will emphasize homeomorphisms rather than isometries.

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