By John B. Conway

This textbook in aspect set topology is aimed toward an upper-undergraduate viewers. Its mild velocity might be important to scholars who're nonetheless studying to jot down proofs. necessities contain calculus and a minimum of one semester of research, the place the coed has been correctly uncovered to the tips of easy 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 path. Metric areas are one of many extra widely used topological areas utilized in different components and are consequently brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the strategy of the e-book firstly the actual and paintings towards the extra normal. 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 capabilities, culminating in a improvement of paracompact areas.

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**Example text**

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. Thus, for each n ≥ 1, Fn = {xk : k ≥ n} contains all ∞ its limit points and is therefore closed.

2. 7. If (X, d) is a metric space and f and g are continuous functions from X into R, then f +g : X → R and f g : X → R are continuous, where (f + g)(x) = f (x) + g(x) and (f g)(x) = f (x)g(x) for all x in X. If f (x) ̸= 0 for all x in X, then f −1 = 1/f : X → R defined by f −1 (x) = 1/f (x) = [f (x)]−1 is a continuous function. 18 1. Metric Spaces The last proposition is a way to combine continuous functions to obtain another continuous function. Here is another. Recall that if f : X → Z and g : Z → W are functions, then the composition of f and g is the function g ◦ f : X → W defined by g ◦ f (x) = g(f (x)).

Thus, the assumption there that G is open is essential. We close this section with a circumspective remark. We did not use sequences in this section. Connectedness is one of the only properties of a metric space I know whose examination never uses the concept of a convergent sequence. 2. 2, that a subset E of R is an interval if and only if when a, b ∈ E and a < b, then c ∈ E whenever a < c < b? (3) If (X, d) is connected and f : X → R is a continuous function such that |f (x)| = 1 for all x in X, show that f must be constant.