72 0 obj endobj (4.1. endobj 40 0 obj << /S /GoTo /D (subsection.3.1) >> endobj << /S /GoTo /D (section.3) >> Continuity improved: uniform continuity) << /S /GoTo /D (subsection.3.2) >> 3 0 obj stream 108 0 obj endobj << /S /GoTo /D (section.6) >> 152 0 obj << endobj << /S /GoTo /D (subsection.2.2) >> << /S /GoTo /D (section.10) >> << /S /GoTo /D (subsection.12.1) >> endobj 68 0 obj Compact spaces) endobj Then there exists a sequence (x n) n2N Sconverging to x. endobj 75 0 obj If has discrete metric, 2. << /S /GoTo /D (section.2) >> endobj 12 0 obj 115 0 obj (3.1. endobj (REFERENCES) 92 0 obj (3. (7.1. endobj endobj (12.2. 112 0 obj endobj Obviously, this sequence is a Cauchy sequence, and, since Sis complete, it converges to some x~ 2S. endobj << /S /GoTo /D (subsection.8.1) >> endobj (7. endobj One-point compactification of topological spaces) Proposition 1.1. (6.2. endobj endobj Let (X;d) be a complete metric space and S X. Theorem 4. Dealing with topological spaces72 11.1. 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. 51 0 obj Connected spaces) The purpose of this chapter is to introduce metric spaces and give some definitions and examples. << /S /GoTo /D (section.7) >> << /S /GoTo /D (subsection.2.1) >> endobj What topological spaces can do that metric spaces cannot) endobj Complete spaces) 7 0 obj endobj What topological spaces can do that metric spaces cannot82 12.1. 71 0 obj (10. (4. Properties of open subsets and a bit of set theory) Then (1 n) is a Cauchy sequence which is not convergent in X. Definition 3. endobj Topological spaces) Proof. endobj 28 0 obj 103 0 obj 83 0 obj Properties of complete spaces) 4 0 obj (1. 1. 76 0 obj 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. endobj is a complete metric space iff is closed in Proof. endobj �8Ik���ÄIV�Z��Ӻ�vj��"k����R�1c��Ӡ�4��A�E�aC����:��|1��kk��Л��?�LI��;l|S��r��C\`�L,c��k�tu��d�ν�в{�X���ot�$��&�h��I�e�m6�M�Z��}�4[��C�qU*r��o!��N�vV�l]�����. 96 0 obj 52 0 obj << /S /GoTo /D (section.4) >> Interlude) 8 0 obj Path-connected spaces) 35 0 obj endobj Open subsets) /Length 944 endobj A metric space (X,d) is said to be complete if every Cauchy sequence in X converges (to a point in X). Properties of complete spaces58 8.2. %PDF-1.4 Then Sis completeifandonlyifSisclosed. endobj 63 0 obj << /S /GoTo /D (section.9) >> 11 0 obj endobj << /S /GoTo /D (subsection.11.2) >> Quotient topological spaces) << /S /GoTo /D (subsection.6.1) >> (8.2. (11. << /S /GoTo /D (subsection.4.1) >> 100 0 obj (9. endobj endobj Note that each x n is an irrational number (i.e., x n 2Qc) and that fx ngconverges to 0. 67 0 obj (6.1. Proof. A very basic metric-topological dictionary) Complete spaces54 8.1. endobj << /S /GoTo /D (section.8) >> endobj endobj endobj 95 0 obj << /S /GoTo /D (subsection.11.1) >> endobj endobj 91 0 obj Dealing with topological spaces) 88 0 obj From metric spaces to topological spaces75 11.2. 23 0 obj Introduction) 24 0 obj (12.1. Topological spaces68 10.1. 20 0 obj /Filter /FlateDecode 60 0 obj Definition. The completion of a metric space) endobj 56 0 obj << /S /GoTo /D (subsection.6.2) >> (8. << /S /GoTo /D (subsection.12.2) >> Connected components) (2. Examples include the real numbers with the usual metric, the complex numbers, finite-dimensional real and complex vector spaces, the space of square-integrable functions on the unit interval L^2([0,1]), and the p-adic numbers. 19 0 obj << /S /GoTo /D (subsection.3.3) >> 43 0 obj 15 0 obj >> xڕW�r�8��+X��^1;w��x*gWMW�̂`�V���X��W��0��Q6��=瞫����n��;��˫O78� �(����C � !�@��-W���l�8��� z�ž,�����zzg��׳��˿[email protected]�n�X�P�+J ��$d�Y��$��7�6 0����{���6 �!�kϾ<6F{�H�A�x�!��5�I �\�B��L%OHz?1�>>d~���5�Z�_�.Df�wi��)0���}����L��`�C\���{���휹WE�W���־x�U��3�A?q�\}��]�͈� ����5��q��-��\��ؘ�@��e��,�!�r��뀍�SJ v�˲�F�@�4ϑO��T61K��Y��}�S��7\��D!L*%��ĝ��Jx��2�Cğe��0��ԥpC3�#W�H���Te��R��&����_��ufѝ�?U�U���F��A���=��1�y��U��끍z%��r�G��I-9Ɲ����&���\ނ�-sdK�>�z�9aJ�O�3:�B��&�߀ � B5�� ���Nec{�:j�п���w�:�U�'J)^L%o]����E�M��뻶�_���"W�X/�gj_� Equivalent metrics) 47 0 obj 59 0 obj endobj << /S /GoTo /D (section.11) >> 79 0 obj 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. Homeomorphisms of metric spaces and open maps) Convergence of sequences in metric spaces) For example if I change real numbers into rational number with usual metric ( absolute value ) it would be incomplete. << /S /GoTo /D (section.1) >> If is the real line with usual metric, , then Remarks. For example, let X = (0,1]. (3.2. Metric spaces: basic definitions) (11.2. 80 0 obj The completion of a metric space61 9. endobj Product spaces) 119 0 obj << /S /GoTo /D (subsection.10.1) >> endobj << /S /GoTo /D (section.5) >> Interlude II66 10. endobj �Q4b�u�a��'0U7?�OϤ�H�$6E�BG Assume that is closed in Let be a Cauchy sequence, Since is complete, But is closed, so On the other hand, let be complete, and let be a limit point of so (in ), . Interlude II) (2.2. 107 0 obj (11.1. Normed real vector spaces) Continuous functions between metric spaces) (10.1. On the other hand if have a some kind of metric on some space it would be incomplete though. One measures distance on the line R by: The distance from a to b is |a - b|. 39 0 obj << /S /GoTo /D (subsection.7.1) >> A metric space is complete if every cauchy sequence is convergent. A closed subset of a complete metric space is a complete sub- space. 1.5 Theorem. endobj << /S /GoTo /D [121 0 R /Fit ] >> 2. endobj %���� endobj 16 0 obj 120 0 obj 99 0 obj 31 0 obj endobj endobj endobj << /S /GoTo /D (subsection.8.2) >> endobj (8.1. endobj From metric spaces to topological spaces) 27 0 obj We do not develop their theory in detail, and we leave the verifications and proofs as an exercise. 36 0 obj (5. << /pgfprgb [/Pattern /DeviceRGB] >> To make space incomplete either i can change the metric or the ambient space. 111 0 obj << /S /GoTo /D (section.12) >> 84 0 obj Definition and examples of metric spaces. 87 0 obj endobj Set theory revisited) 104 0 obj 44 0 obj endobj (3.3. endobj endobj endobj Completion of a metric space A metric space need not be complete. (12. Thus, fx ngconverges in R (i.e., to an element of R). 116 0 obj is called open if is called ... Let be a complete metric space, . endobj Set theory revisited70 11. 48 0 obj A complete metric space is a metric space in which every Cauchy sequence is convergent. 32 0 obj endobj (=)) Let x2S. 55 0 obj (6. endobj endobj Examples. A very basic metric-topological dictionary78 12. << /S /GoTo /D (section*.1) >> 64 0 obj (2.1. Cauchy Sequences and Complete Metric Spaces Let’s rst consider two examples of convergent sequences in R: Example 1: Let x n = 1 n p 2 for each n2N. Let S be a closed subspace of a complete metric space X. endobj