43. Proof. 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. Set theory revisited70 11. 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. If a metric space Xis not complete, one can construct its completion Xb as follows. Thus, U is a union of open balls and the proof is complete. 252 Appendix A. Then {x n} converges itself. 1 Initial Construction This construction will rely heavily on sequences of elements from the metric space (E;d). 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). This abstracts the Bolzano{Weierstrass property; indeed, the Bolzano{Weierstrass theorem states that closed bounded subsets of the real line are sequentially compact. Example 4 revisited: Rn with the Euclidean norm is a Banach space. 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. From metric spaces to topological spaces75 11.2. Theorem 1.3 – Limits are unique The limit of a sequence in a metric space is unique. 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. Dealing with topological spaces72 11.1. De nition: A complete normed vector space is called a Banach space. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. 8/37 . A very basic metric-topological dictionary78 12. Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. Let an element ˘of Xb consist of an equivalence class of Cauchy 251. 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. Ark2: Complete and compact spaces MAT2400 — spring 2012 continuous. Topological spaces68 10.1. What topological spaces can do that metric spaces cannot82 12.1. Properties of complete spaces58 8.2. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. In other words, no sequence may converge to two diﬀerent limits. Metric Spaces, Topological Spaces, and Compactness sequences in X;where we say (x ) ˘ (y ) provided d(x ;y ) ! Complete Metric Spaces 6 Theorem 43.5. The completion of a metric space61 9. Interlude II66 10. Complete spaces54 8.1.