Quotient topology 52 6.2. ric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. Let ϵ>0 be given. (iii) A and B are both closed sets. Note that iff If then so Thus On the other hand, let . 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� A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. �fWx��~ ]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���. 256 0 obj
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Therefore, it becomes completely ineffective when the space is discrete (consists of isolated points) However, these discrete metric spaces are not always identical (e.g., Z and Z 2). iff is closed. Exercise 11 ProveTheorem9.6. Examples. 1 0 obj
Convergence of mappings. Topology of Metric Spaces S. Kumaresan. Definition: If X is a topological space and FX⊂ , then F is said to be closed if FXFc = ∼ is open. Covering spaces 87 10. It is often referred to as an "open -neighbourhood" or "open … Continuous Functions 12 8.1. METRIC SPACES AND TOPOLOGY Denition 2.1.24. %����
A subset S of the set X is open in the metric space (X;d), if for every x2S there is an x>0 such that the x neighbourhood of xis contained in S. That is, for every x2S; if y2X and d(y;x) < x, then y2S. If B is a base for τ, then τ can be recovered by considering all possible unions of elements of B. stream
Details of where to hand in, how the work will be assessed, etc., can be found in the FAQ on the course Learn page. 1.1 Metric Spaces Deﬁnition 1.1.1. That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. Fibre products and amalgamated sums 59 6.3. to the subspace topology). Homotopy 74 8. It consists of all subsets of Xwhich are open in X. 4.2 Theorem. 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. The function f is called continuous if, for all x 0 2 X , it is continuous at x 0. If a pseudometric space is not a metric spaceÐ\ß.Ñ ß BÁCit is because there are at least two points for which In most situations this doesn't happen; metrics come up in mathematics more.ÐBßCÑœ!Þ often than pseudometrics. De nition A1.3 Let Xbe a metric space, let x2X, and let ">0. then B is called a base for the topology τ. set topology has its main value as a language for doing ‘continuous geome- try’; I believe it is important that the subject be presented to the student in this way, rather than as a … METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under-standing and ability to make use of properties of U U1. The ﬁrst goal of this course is then to deﬁne metric spaces and continuous functions between metric spaces. <>
General Topology 1 Metric and topological spaces The deadline for handing this work in is 1pm on Monday 29 September 2014. For a metric space (X;d) the (metric) ball centered at x2Xwith radius r>0 is the set B(x;r) = fy2Xjd(x;y)

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