SET THEORY AND METRIC SPACES . /Name/F1 6 0 obj << Trent University Library Donation. endobj /Type/Font Proof. - 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. 299.8 731.4 444.1 444.1 626.9 624.5 625.7 600.8 678 561 534.9 626.9 663.1 258.8 442.9 �J�D(��'nҦ��V�i��e��T"�J��E�;V2�Í(�����s. 9 0 obj The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. Note that iff If then so Thus On the other hand, let . This means that ∅is open in X. 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 ["Ư8�T�x\$�Pn�6i�R����"N .^gwfvwb@0!`|���u��x�1{�_��>�N/)`K �M_�0Q��\$��^ծ5(b��ܕ�F�:3D3�L�iz@Me�e��PY4*�6�U���EO�Ǫ��>��\5�d׀J��O1������u�2w�i�D|�}�sX�*0�"4�X�^�R,�W;[�U����g�Ү�%���R�(��C�G�+]��s� /Encoding 10 0 R >> endobj /F1 9 0 R /Type/Font Theorem 2.6. 0000000593 00000 n endobj /Widths[299.8 470.2 783.7 470.2 783.7 712.1 261.2 365.7 365.7 470.2 731.4 261.2 313.5 (Alternative characterization of the closure). >> space is sometimes called a Polish space. It is assumed that measure theory and metric spaces are already known to the reader. 19 0 obj xڵR�N�0��>! stream 777.8 500 861.1 972.2 777.8 238.9 500] 693 576 537 694 738 324 444 611 520 866 713 731 558 731 646 556 597 694 618 928 600 /BaseFont/KQIRGL+CMSS10 << >> /Type/Encoding 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 508.8 783.7 731.4 572.3 626.9 663.1 /Length 1617 329 833 367 367 500 547 484 278 500 367 404 442 867 867 868 486 639 639 639 639 639 >> Here are to be found only basic issues on continuity and measurability of set-valued maps. 0 0 0 0 0 0 541.7 833.3 777.8 611.1 666.7 708.3 722.2 777.8 722.2 777.8 0 0 722.2 /Filter[/FlateDecode] Theorem 1.2 – Main facts about open sets 1 If X is a metric space, then both ∅and X are open in X. We need a lemma from topology. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. iff ( is a limit point of ). /FirstChar 33 483.9 431.6 640.6 431.6 431.6 408.3 470.2 940.4 470.2 470.2 470.2 0 0 0 0 0 0 0 0 Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function Because of the gener-ality of this theory, it is useful to start out with a discussion of set theory itself. /FirstChar 33 14/Zcaron/zcaron/caron/dotlessi/dotlessj/ff/ffi/ffl/notequal/infinity/lessequal/greaterequal/partialdiff/summation/product/pi/grave/quotesingle/space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde At the same time the top-ics on topological spaces are taken up as long as they are necessary for the discussions on set-valued maps. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. 238.9 794.4 516.7 500 516.7 516.7 341.7 383.3 361.1 516.7 461.1 683.3 461.1 461.1 NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. /Length 35 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 288.9 500 277.8 277.8 480.6 516.7 444.4 516.7 444.4 305.6 500 516.7 238.9 266.7 488.9 The closure of a set is defined as Theorem. Set theory and metric spaces. 0000011079 00000 n 0000001189 00000 n 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. << 2 Arbitrary unions of open sets are open. 15 0 obj 0000000648 00000 n �4�l:5v!i�UM5v( �h:�6����R,.�i�e��A�����������G�������Y��eV��E�B#�w[L�[�I�. f1.3ye2/f1.3yk3 algebra and analysis part 1: analysis. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. �tL ���@�Th��dH"�8�E�� ���8��L�S�64�0���>@�C��:J�0U� ��*�h��RP��&��1ߠ��a�Ɏ���M��WD��PA�������A��0�WKKAi7��}>�[��ު�T�\\$�I%^����N�L�K�*_�����mun����_���}R�����/HR��*_���u��*뮺�K�V��Z������_J���פ����C���n�����l%UꐯI�k�W�:��V!k�eX�~��V��U��q����T���-�J]UZ_Yu�0�Ԏ��3�#�V�UT�M��R��a.�C�m Hence, only a review has been made of metric spaces. IN COLLECTIONS. 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���������RR�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 0000008912 00000 n /Name/F3 >> Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. /BaseFont/IPVJTN+CharterBT-Roman 10 0 obj 0 0 0 0 0 0 0 500 170 278 338 331 745 556 852 704 201 417 417 500 833 278 319 278 If (X;d) is a complete separable metric space, then every nite Borel measure on Xis tight. endobj 736.1 638.9 736.1 645.8 555.6 680.6 687.5 666.7 944.4 666.7 666.7 611.1 288.9 500 Examples of Metric Spaces …