In this paper, we introduce a new contraction condition which is assumed to hold for comparable elements of a subset of whole space. ��h��[��b�k(�t�0ȅ/�:")f(�[S�b@���R8=�����BVd�O�v���4vţjvI�_�~���ݼ1�V�ūFZ�WJkw�X�� <> 9 0 obj <>>> Assume that is not sequentially compact. One measures distance on the line R by: The distance from a to b is |a - b|. endobj ~"���K:��d�N��)������� ����˙��XoQV4���뫻���FUs5X��K�JV�@����U�*_����ւpze}{��ݑ����>��n��Gн���3`�݁v��S�����M�j���햝��ʬ*�p�O���]�����X�Ej�����?a��O��Z�X�T�=��8��~��� #�$ t|�� 1. is compact. 501, SPRING 2017 JACEKPOLEWCZAK Example 1 Let Qbe a the set of all rational numbers with the metric given by d(x,y) = |x −y|, for x,y ∈ Q. Note that compactness depends only on the topology, while … i�Z����Ť���5HO������olK�@�1�6�QJ�V0�B�w�#�Ш�"�K=;�Q8���Ͼ�&4�T����4Z�薥�½�����j��у�i�Ʃ��iRߐ�"bjZ� ��_������_��ؑ��>ܮ6Ʈ����_v�~�lȖQ��kkW���ِ���W0��@mk�XF���T��Շ뿮�I؆�ڕ� Cj��- �u��j;���mR�3�R�e!�V��bs1�'�67�Sڄ�;��JiY���ִ��E��!�l��Ԝ�4�P[՚��"�ش�U=�t��5�U�_:|��Q�9"�����9�#���" ��H�ڙ�×[��q9����ȫJ%_�k�˓�������)��{���瘏�h ���킋����.��H0��"�8�Cɜt�"�Ki����.R��r ������a�$"�#�B�$KcE]Is��C��d)bN�4����x2t�>�jAJ���x24^��W�9L�,)^5iY��s�KJ���,%�"�5���2�>�.7fQ� 3!�t�*�"D��j�z�H����K�Q�ƫ'8G���\N:|d*Zn~�a�>F��t���eH�y�b@�D���� �ߜ Q�������F/�]X!�;��o�X�L���%����%0��+��f����k4ؾ�۞v��,|ŷZ���[�1�_���I�Â�y;\�Qѓ��Џ�`��%��Kz�Y>���5��p�m����ٶ ��vCa�� �;�m��C��#��;�u�9�_��`��p�r�`4 2. Consider the sequence {x n} of rational numbers such that x1 = 1 and x n+1 = 2 1+x n 2+x n, for n ≥ 2. What is its completion, ((0;1) ;d))? %PDF-1.4 x��]�o7�7��a�m����E` ���=\�]�asZe+ˉ4Iv���*�H�i�����Hd[c�?Y�,~�*�ƇU���n��j�Yiۄv��}��/����j���V_��o���b�޾]��x���phC���>�~��?h��F�Շ�ׯ�J�z�*:��v����W�1ڬTcc�_}���K���?^����b{�������߸����֟7�>j6����_]������oi�I�CJML+tc�Zq�g�qh�hl�yl����0L���4�f�WH� Theorem. Any incomplete space. ]F�)����7�'o|�a���@��#��g20���3�A�g2ꤟ�"��a0{�/&^�~��=��te�M����H�.ֹE���+�Q[Cf������\�B�Y:�@D�⪏+��e�����ň���A��)"��X=��ȫF�>B�Ley'WJc��U��*@��4��jɌ�iT�u�Nc���դ� ��|���9�J�+�x^��c�j¿�TV�[�•��H"�YZ�QW�|������3�����>�3��j�DK~";����ۧUʇN7��;��`�AF���q"�َ�#�G�6_}��sq��H��p{}ҙ�o� ��_��pe��Q�$|�P�u�Չ��IxP�*��\���k�g˖R3�{�t���A�+�i|y�[�ڊLթ���:u���D��Z�n/�j��Y�1����c+�������u[U��!-��ed�Z��G���. A metric space is compact if and only if it is complete and totally bounded. �?��Ԃ{�8B���x��W�MZ?f���F��7��_�ޮ�w��7o�y��И�j�qj�Lha8�j�/� /\;7 �3p,v <> Proof: Exercise. Some important properties of this idea are abstracted into: Definition A metric space is a set X together with a function d (called a metric or "distance function") which assigns a real number d(x, y) to every pair x, y X satisfying the properties (or axioms): d(x, y) 0 and d(x, y) = 0 x = y, d(x, y) = d(y, x), d(x, … endobj %PDF-1.5 stream Turns out, these three definitions are essentially equivalent. 2 0 obj 3 0 obj 3. is complete and totally bounded. Definition and examples of metric spaces. The following properties of a metric space are equivalent: Proof. x�jt�[� ��W��ƭ?�Ͻ����+v�ׁG#���|�x39d>�4�F[�M� a��EV�4�ǟ�����i����hv]N��aV endobj %�쏢 �fWx��~ Non-examples. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 594.6 843.24] /Contents 4 0 R/Group<>/Tabs/S/StructParents 1>> <> x��\Y�\�u�_�����S%�=0�H�=�2��>�w����e�I��j�����;��S���7�������u���t���ۍ���Iݿ\q������z���|�b��moRw�z���ӝ��ʇ�o�1Y�C���|��n:���bP[u�~�ۛ�[e�����]ʽf�k�wٛ��7�fӻh���ӽs6����=�$������ކzո�Vi�V���^����ƫ0��t��q�`uk�U�m?�v!fF�|�dǥ�'�)��-��^�X��L2�J㚯�~����I�n�g�sw�*�8��.L�Z��v�JPYOv|������ Example 9: The open unit interval (0;1) in R, with the usual metric, is an incomplete metric space. stream 1. 2. is sequentially compact. EXAMPLES OF INCOMPLETE METRIC SPACES MATH. This is known as the Heine–Borel theorem. %���� 1 0 obj Every closed subset of a compact space is itself compact. Mn�qn�:�֤���u6� 86��E1��N�@����{0�����S��;nm����==7�2�N�Or�ԱL�o�����UGc%;�p�{�qgx�i2ը|����ygI�I[K��A�%�ň��9K# ��D���6�:!�F�ڪ�*��gD3���R���QnQH��txlc�4�꽥�ƒ�� ��W p��i�x�A�r�ѵTZ��X��i��Y����D�a��9�9�A�p�����3��0>�A.;o;�X��7U9�x��. ~��Ϻ��� ��n��X[kp�9g�����@@��5���F+�n���`���Y�����M�!��+��C|{UP��AAzM#��;����b������e���rR�?�:����/�v��5���T]���Oa�Mfj��>"e ��l�O�m�D������i��Z&d������C�v{3����kC�R����#�f[;�X�)xDr�B"%���m��3.t�Z�B ��O���)�6$BLS��rp �c�59I�1�g3oi�����k�湘�Gn_�X7��C�:kL�޾k����G���c���p� ������Q��i� ���w2թa<0�`�Yz?i���'�������r����]��S. 4 0 obj In an ordered metric space, completeness implies -completeness. Examples of compact metric spaces include the closed interval [,] with the absolute value metric, all metric spaces with finitely many points, and the Cantor set. An ordered metric space is said to be -complete, if every increasing Cauchy sequence in converges in . + Theorem: A subset of a complete metric space is itself a complete metric space if and only if it is closed. Any unbounded subset of any metric space.

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