# complete metric space

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November 29th, 2020

Theorem 1. Completeness is not a topological property, i.e. Completeness is not a topological property, that is, there are metric spaces which are homeomorphic as topological spaces, one being complete and the other not. Denote by C [X] the collection of all Cauchy sequences in X. Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. Hence, we will have to make some adjustments to this initial construction, which we shall undertake in the following sections. Completion of a Metric Space Definition. Every metric space has a completion. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. Let (X;d X) be a complete metric space and Y be a subset of X:Then (Y;d Y) is complete if and only if Y is a closed subset of X: Proof. one can’t infer whether a metric space is complete just by looking at the underlying topological space. Proof. Proof. The resulting space will be denoted by Xand will be called the completion of … This is left to the reader as an exercise. Let (X, d) be a metric space. The Completion of a Metric Space Let (X;d) be a metric space. A metric space is called complete if every Cauchy sequence converges to a limit. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. A completion of a metric space (X, d) is a pair consisting of a complete metric space (X *, d *) and an isometry ϕ: X → X * such that ϕ [X] is dense in X *. Already know: with the usual metric is a complete space. The goal of these notes is to construct a complete metric space which contains X as a subspace and which is the \smallest" space with respect to these two properties. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Proposition 1.1. A metric space (X, d) is said to be complete if every Cauchy sequence in X converges. The hope with this initial construction is that (C[E];D) is a complete metric space, but, as will be seen in part (v) of Exercise 1.2, Dfails to even be a metric. Theorem. Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). For example, consider the real line \$\mathbb{R}\$ and the open unit interval \$(-1,1)\$, each with the usual metric. B) of R2 is not a complete metric space. with the uniform metric is complete. This proposition allows us to construct many examples of metric spaces which are not complete. Since is a complete space, the sequence has a limit.