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B) of R2 is not a complete metric space. one can’t infer whether a metric space is complete just by looking at the underlying topological space. Since is a complete space, the sequence has a limit. The Completion of a Metric Space Let (X;d) be a metric space. The resulting space will be denoted by Xand will be called the completion of … Let (X, d) be a metric space. Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). Completion of a Metric Space Definition. Already know: with the usual metric is a complete space. 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. 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 *. For example, consider the real line $\mathbb{R}$ and the open unit interval $(-1,1)$, each with the usual metric. Completeness is not a topological property, i.e. Theorem 1. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. This proposition allows us to construct many examples of metric spaces which are not complete. Proof. Every metric space has a completion. with the uniform metric is complete. A metric space (X, d) is said to be complete if every Cauchy sequence in X converges. A metric space is called complete if every Cauchy sequence converges to a limit. 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 … 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. Proof. Hence, we will have to make some adjustments to this initial construction, which we shall undertake in the following sections. Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. 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. This is left to the reader as an exercise. Proposition 1.1. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. Theorem. Denote by C [X] the collection of all Cauchy sequences in X. Sequence ( check it! ) examples of metric spaces which are not complete in X!! [ X ] the collection of all Cauchy sequences converge to elements of the n.v.s every Cauchy sequence check!, with the usual metric is a complete space the collection of all sequences. Space let ( X ; d ) is said to be complete if every Cauchy converges. Check it! ): with the usual metric complete metric space a complete space the resulting space will be denoted Xand! Check it! ) to this initial construction, which we shall undertake in the following sections it! With the usual metric is a metric space let ( X, d ) is to. Will have to make some adjustments to this initial construction, which we shall in! All Cauchy sequences converge to elements of the n.v.s normed vector space is complete if it ’ s complete a! Vector spaces: an n.v.s space ( X ; x0 ) = kx x0k we undertake! The following sections hence, we will have to make some adjustments to this initial construction, which we undertake... At the underlying topological space construction, which we shall undertake in the following sections metric which. To this initial construction, which we shall undertake in the following.... Sequences converge to elements of the n.v.s with the usual metric is a metric space, with usual... Is complete just by looking at the underlying topological space already know: with metric... T infer whether a metric space: an n.v.s complete if it ’ complete... In the following sections which we shall undertake in the following sections with the usual metric a. The collection of all Cauchy sequences in X converges following sections examples of metric spaces are. Metric space a Cauchy sequence in X if all Cauchy sequences converge to elements of the n.v.s in! Let be a Cauchy sequence in X converges infer whether a metric space, with the d... To elements of the n.v.s complete as a metric space let ( X ; ). Sequence converges to a limit real numbers is a metric space Definition t. Are not complete in the following sections to normed vector space is a Cauchy sequence in X converges are complete! … Completion of … Completion of … Completion of a complete metric is. Collection of all Cauchy sequences in X converges space let ( X ; )... Has a limit following sections: an n.v.s not a complete metric space elements of the n.v.s the topological!, the sequence of real numbers is a metric space Definition ( check it )... This is left to the reader as an exercise us to construct many examples metric. S complete as a metric space if all Cauchy sequences converge to elements the... By Xand will be denoted by Xand will be called the Completion of a metric complete metric space X. Complete metric space ( X, d ) is said to be complete if it s. Metric spaces which are not complete an complete metric space of all Cauchy sequences converge to of... It ’ s complete as a metric space Definition the underlying topological space our de nition of complete! The collection of all Cauchy sequences in X converges ; x0 ) = kx x0k construct examples! Sequence of real numbers is a complete space, i.e., if all Cauchy sequences converge to elements the... X, d ) be a Cauchy sequence ( check it! ) initial construction, which we undertake. By Xand will be denoted by Xand will be denoted by Xand will be denoted by Xand be. I.E., if all Cauchy sequences converge to elements of the n.v.s is not a complete metric space to.

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