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Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. If (Z,d Z) is a third metric space, show that a function f: Z → X × Y is continuous at z ∈ Z if and only if the two compositions p X f and p Y f are. Set theory revisited70 11. Let an element ˘of Xb consist of an equivalence class of Cauchy 251. In other words, no sequence may converge to two diﬀerent limits. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. Ark2: Complete and compact spaces MAT2400 — spring 2012 continuous. Proof. Properties of complete spaces58 8.2. 8/37 . Interlude II66 10. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Then {x n} converges itself. De nition: A complete normed vector space is called a Banach space. From metric spaces to topological spaces75 11.2. The completion of a metric space61 9. 1 Initial Construction This construction will rely heavily on sequences of elements from the metric space (E;d). Metric Spaces, Topological Spaces, and Compactness sequences in X;where we say (x ) ˘ (y ) provided d(x ;y ) ! A metric space X is said to be sequentially compact if every sequence (xn)1 n=1 of points in X has a convergent subsequence. A very basic metric-topological dictionary78 12. Complete Metric Spaces 6 Theorem 43.5. If a metric space Xis not complete, one can construct its completion Xb as follows. Deﬁnition – Complete metric space A metric space (X,d) is called complete if every Cauchy sequence of points of X actually converges to a point of X. Theorem 1.13 – Cauchy sequence with convergent subsequence Suppose (X,d) is a metric space and let {x n} be a Cauchy sequence in X that has a convergent subsequence. Theorem 1.3 – Limits are unique The limit of a sequence in a metric space is unique. Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. If the space Y is complete in the metric d, then the space YJ is complete in the uniform metric ρ corresponding to d. Note. This abstracts the Bolzano{Weierstrass property; indeed, the Bolzano{Weierstrass theorem states that closed bounded subsets of the real line are sequentially compact. 252 Appendix A. That is, we will construct a new metric space, (E;d), which is complete and contains our original space Ein some way (to be made precise later). Complete spaces54 8.1. Example 4 revisited: Rn with the Euclidean norm is a Banach space. Thus, U is a union of open balls and the proof is complete. Topological spaces68 10.1. 43. Dealing with topological spaces72 11.1. What topological spaces can do that metric spaces cannot82 12.1. Spaces cannot82 12.1 nition: a complete metric space ( E ; d.... 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