set theory and metric spaces pdf
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2 Arbitrary unions of open sets are open. 481 556 556 556 556 556 556 556 556 556 556 319 319 833 833 833 486 942 639 604 632 A�m->+N�����������iXa.��JתmLW�HAն����k��[��i�&�C[UM{MS CUTL&5�aC-E; ��!3!����b#A�k�%�/�aPD��0�(�+T´�0�#������������p�}��/ZZ��������������������������������������������������������p�۱������������������������������������������������������������������堥G�(�dK�6-DuS�%A��e()�q�#z�0�t ���9�@�Q��#PC�;V2�1 ����p@�x4 �4�g 4C/�"�`�� �a4��[�>�p��L:֝��;h �� ����&$K��eX0����N!����B d4��$E>��A�A�@�dC�I4ȇ��Ma��I0�A�� ��v�ݥzkvݧzi^���'ۤ�������{����V�=�}�W����������{�������K��WI����������n���*�C3���������RR�lt����匿z�_���W���z��E�����=R�/��~4��?����׾� {�7�����#8.Ã#����� �������[�zK��?oZJ�[�0� ���7��=� �����-�xo���S��|�U��܋=�]�nE�᷿�����t�]m�n��ڧ�������ް����&O�z����ԧˠ�KC�o#�W�� w~��ݦ�J�N�n�ۿwJ�M���U��a ���1 4�%wI��nøMnp�P@� !PiD1��@f��`D0�0�1d1�0҄!Pc0@˃H+��a� � �4݈-�J�.�U���S����i�4 /Name/F1 0000007831 00000 n /Font 17 0 R >> /Filter[/FlateDecode] endstream It is assumed that measure theory and metric spaces are already known to the reader. Set theory and metric spaces. /Type/Font << /Filter[/FlateDecode] >> /F2 13 0 R 470.2] xڵR�N�0��>! 0000000898 00000 n 20 0 obj << /Linearized 1 /L 40359 /H [ 736 182 ] /O 23 /E 12278 /N 4 /T 39915 >> endobj xref 20 16 0000000016 00000 n 280 528 568 539 539 539 539 539 833 539 569 569 569 569 495 551 495] /Widths[319.4 500 833.3 500 833.3 758.3 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 endobj 434.7 500 1000 500 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 obj 0000000918 00000 n 0000001834 00000 n /Subtype/Type1 /FirstChar 1 /Name/F2 stream >> endobj trailer << /Size 36 /Prev 39905 /Info 19 0 R /Root 21 0 R >> startxref 0 %%EOF 21 0 obj << /Type /Catalog /Pages 22 0 R >> endobj 22 0 obj << /Type /Pages /Kids [ 23 0 R 1 0 R 7 0 R 13 0 R ] /Count 4 >> endobj 34 0 obj << /Length 35 0 R /S 56 /Filter /FlateDecode >> stream If (X;d) is a complete separable metric space, then every nite Borel measure on Xis tight. f1.3ye2/f1.3yk3 algebra and analysis part 1: analysis. 777.8 500 861.1 972.2 777.8 238.9 500] endobj endobj space will be a set Xwith some additional structure. 400 245 817 0 0 586 0 338 556 556 606 556 500 500 500 900 380 442 833 319 900 500 �S���Q2��q�պ���!�ѭܴ��q �ժ͋���3y��g1�$ X���UH�u��[[�}`��P���E=��������3�&��n��qL�"#D����~�e��k /Type/Font Books to Borrow. The closure of a set is defined as Theorem. /Encoding 10 0 R !C�0�^�">_��h ���RuzTA��Rza|E*Kƫ�R�K�HG� In most cases, the proofs 10 0 obj Hence, only a review has been made of metric spaces. Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function << 0000000736 00000 n 0000009992 00000 n Lemma 2.7. �4��������c֋%���3O,�Z�ͩ���7���Y�YƢ}�:/����t�o���.��j�����+���Jp�B� ��áz)�c�{uax�;��#�P��3z�����>���9Ú��A8��A�����H�t�Yة;�A��n�t��1�7V�BL��zƘ�E0�Ę0s�'�C��ƫ5?�= ������]���i�3�(mpD�?��.��=����\t�3�gH��= ޷MS�T��0�t��(J�D��]���Kl�� ��<>�({;����L@ endstream endobj 27 0 obj 4612 endobj 28 0 obj << /Type /XObject /Subtype /Image /Name /im1 /Length 27 0 R /Width 1614 /Height 2598 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /EndOfLine false /EncodedByteAlign false /Columns 1614 /EndOfBlock true >> >> stream (Alternative characterization of the closure). 25 0 obj << We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. 650.6 508.8 819.8 663.1 692.8 599.6 692.8 606.4 522.4 640.6 643.8 624.5 885.7 624.5 /ProcSet[/PDF/Text/ImageC] %PDF-1.2 %���� 0000001189 00000 n endobj 586 586 421 481 421 1000 500 201 507 539 446 565 491 321 523 564 280 266 517 282 Proof. /Differences[1/dotaccent/fi/fl/fraction/hungarumlaut/Lslash/lslash/ogonek/ring 11/breve/minus endobj 0000001812 00000 n space is sometimes called a Polish space. ��� T /Filter[/FlateDecode] CONTENTS COMPLETENESS, SEPARABILITY, AND COMPACTNESS 84 5.7 5.2 5.3 Completeness Separability Compactness 84 94 98 6 ADDITIONAL TOPICS 106 6.1 Product Spaces 106 6.2 A Fixed-point Theorem 108 6.3 Category 111 APPENDIXES 115 7. To show that X is x�c```c``z���� �� �� 6P���H��20H�ҁ�Hj����A�O`h����(,ˢƢ¢̢Ţ�� ��� endstream endobj 35 0 obj 76 endobj 23 0 obj << /Type /Page /Parent 22 0 R /MediaBox [ 0 0 387 623 ] /Resources 24 0 R /Contents 26 0 R >> endobj 24 0 obj << /ProcSet [ /PDF /Text /ImageB ] /Font << /F4 29 0 R /F0 30 0 R /F1 31 0 R /F6 32 0 R /F2 33 0 R >> /XObject << /im1 28 0 R >> >> endobj 25 0 obj 522 endobj 26 0 obj << /Length 25 0 R /Filter /FlateDecode >> stream 2K�w4�6�-��A����*�'�lW�MT���ٿ�Ak&��Z�5�K7�p}��]�����N0�S�3��S1����e�TQ����Y-��� << Examples of Metric Spaces … 693 576 537 694 738 324 444 611 520 866 713 731 558 731 646 556 597 694 618 928 600 17 0 obj /F1 9 0 R Therefore our de nition of a complete metric space applies to normed vector spaces… /Length 1617 Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. 9 0 obj theory of metric spaces lecture notes and exercises NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. This means that ∅is open in X. 470.2 483.9 222.6 248.8 457.7 222.6 745.1 483.9 470.2 483.9 483.9 320.3 360.5 339.6 0000001035 00000 n - by Kaplansky, Irving, 1917-Publication date 1972 Topics Metric spaces, Set theory Publisher Boston: Allyn and Bacon ... 14 day loan required to access EPUB and PDF files.

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