/Filter /FlateDecode << %PDF-1.5 �:��k�,W��QxrSoV�� ���8�Q�y@��]$��2�6{�q�K��M�'$,� %PDF-1.5 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∗. >> Every metric space has a completion. /Filter /FlateDecode Proposition 1.1. 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 *. Then the closure ofMin this larger space is deﬁned to be its completion. x���r��}�Bٗ�31K܉t�Ҵi�i;m�3��fh �YS�CRq���������V�'� �˹ߠtu�JW{�.���ͫ?~�̊�DH�Wo�W,�Y��&�d�7�����[wq)�ZQ��K�uE��5����O�5ņ����w����0�Z�4��e8�����ru�aME��u�6��S��Gc�2�e?���\��tq�����"�&�qw
��������K%U89����[�z�[׆�Uݹ��E��j���U�5�MG����-m N���x��d"QN�0p{q)����ˋ����1�4|���Tµ�C�'�M���_���jKݓ]pd��n���./K�v��L�4E�߸������w5�ꂭ=��U�>\p�� :�'Fdii��M&l�µ�C�u��d���Þ��P��]����1���+��#.��o^���� �+�tb�]i�e�j�{��]���˯Wi"�^=����E�,W߾���a�\�$��Sx�f=j=jNd��CM~B1�0��ug�\(��}fJ�q� W� �J�iă�8��ǯ���� �S�"E��YW���=��J�hӟ)E�@)0�m�l�u�7a%�ٸ�mz.����#h�ҹ-*��~��+ ��. /Length 4396 /Length 3939 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). Completion of a Metric Space Definition. 5 0 obj << C1b8�8�H���� �Ъ��/qs,�e���\����EH 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. stream The Completion of a Metric Space Let (X;d) be a metric space. Completion of a metric space A metric space need not be complete. Theorem 1. The Completion of a Metric Space Brent Nelson Let (E;d) be a metric space, which we will reference throughout. Given an incomplete metric spaceM, we must somehow deﬁne a larger complete space in whichMsits. %���� %���� It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f′ from M′ to N that extends f. The space M' is determined up to isometry by this property (among all complete metric spaces isometrically containing M), and is called the completion of M. Let (X, d) be a metric space. For instance: Bolzano{Weierstrass theorem. Every metric space has a completion. For any metric space M, one can construct a complete metric space M′ (which is also denoted as M), which contains M as a dense subspace. 0, ‘1and ‘2, and to the general construction of completing a metric space. The resulting space will be denoted by Xand will be called the completion of Xwith respect to d. The hard part is that we have nothing to work with except Xitself, and somehow it The procedure is as follows. For example, let B = f(x;y) 2R2: x2 + y2 <1g be the open ball in R2:The metric subspace (B;d B) of R2 is not a complete metric space. The new space is referred to as thecompletionof the space. Proof. Denote by C [X] the collection of all Cauchy sequences in X. Compactness in metric spaces The closed intervals [a;b] of the real line, and more generally the closed bounded subsets of Rn, have some remarkable properties, which I believe you have studied in your course in real analysis. 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. x��[I�۸���P�ĮXV�ˇ̌g��U�*��}�%v�-m�r���y Z���r!���m ��N���+z���۫���Ԅi���7frb���$R䓷�ɻ����� 27 0 obj This is left to the reader as an exercise. stream >> Theorem 1. The purpose of these notes is to guide you through the construction of the \completion" of (E;d). @P���0������y]y��tX�ڎ����6�F)Q��}}d�cE�� }��8ھ�DS�Gj
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