A Course in Point Set Topology by John B. Conway

# A Course in Point Set Topology by John B. Conway

By John B. Conway

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its mild velocity may be worthy to scholars who're nonetheless studying to write down proofs. necessities comprise calculus and at the least one semester of study, the place the coed has been correctly uncovered to the guidelines of uncomplicated set conception akin to subsets, unions, intersections, and capabilities, 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 direction. Metric areas are one of many extra regular topological areas utilized in different parts and are consequently brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the procedure of the e-book to begin with 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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11. A compact metric space is separable. very active in the French government, serving in the French Chamber of Deputies (1924–1936) and as Minister of the Navy (1925–1940). 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 > 0, then choose n such that n−1 < . Thus, there is a point x in Fn ⊆ F with d(x0 , x) < n−1 < , proving that x0 ∈ cl F .

34 1. Metric Spaces Proof. (a) If H is a component of G and x ∈ H, choose r > 0 such that B(x; r) ⊆ G. 7 implies H ∪ B(x; r) is connected. Since this is also a subset of G, it follows that H = H ∪ B(x; r), so B(x; r) ⊆ H, and H is open. Because Rq is separable, there is a countable dense subset D. Now since each component is open, each component contains an element of D and diﬀerent components contain diﬀerent points. ), which is nonsense. (b) Assume that G satisﬁes the stated condition, and let us prove that G is connected.

Fn ∈ F , k=1 Fk = ∅. The following theorem is the main result on compactness in metric spaces. 5. The following statements are equivalent for a closed subset K of a metric space (X, d). (a) K is compact. (b) If F is a collection of closed subsets of K having the FIP, then it holds that F ∈F F = ∅. (c) Every sequence in K has a convergent subsequence. (d) Every inﬁnite subset of K has a limit point. (e) (K, d) is a complete metric space that is totally bounded. Proof. (a) implies (b). Let F be a collection of closed subsets of K having the FIP.