A Course in Point Set Topology (Undergraduate Texts in by John B. Conway
By John B. Conway
This textbook in aspect set topology is geared toward an upper-undergraduate viewers. Its mild velocity may be important to scholars who're nonetheless studying to put in writing proofs. necessities comprise calculus and a minimum of one semester of research, the place the coed has been correctly uncovered to the information of simple set concept akin to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the actual line. Appendices are integrated to bridge the space 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 through the textual content. This additionally conforms to the procedure of the e-book to begin with the actual and paintings towards the extra normal. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of idea, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued features, culminating in a improvement of paracompact areas.
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A) If f, g ∈ U and we define (f + g) : X → R by (f + g)(x) = f (x) + g(x) for all x in X, show that f + g ∈ U. In words, U is a vector space over R. (b) If f, g ∈ U and we define (f g) : X → R by (f g)(x) = f (x)g(x) for all x in X, 24 1. Metric Spaces show by an example that it does not necessarily follow that f g ∈ U. If, however, the functions are also bounded, then f g ∈ U. ] (c) Can you give some conditions under which the quotient of two uniformly continuous functions is uniformly continuous?
2(a) for Rq , but first we need a lemma. 7. If −∞ < a < b < ∞ in R, then [a, b] is compact. Proof. Clearly a closed interval, being a closed subset of a complete metric space, is complete. 5(e)]. But if r > 0, then we can easily find a = x1 < x2 < · · · < xn = b such that xk−1 − xk < r, and from here it is routine to show that [a, b] ⊆ nk=1 B(xk ; r). 8 (Heine5–Borel6 Theorem). A subset of Rq is compact if and only if it is closed and bounded. 5Heinrich Eduard Heine was born in 1821 in Berlin, the eighth of nine children.
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.