## topology of real numbers pdf

o��\$Ɵ���a8��weSӄ����j}��-�ۢ=�X7�M^r�ND'�����`�'�p*i��m�]�[+&�OgG��|]�%��4ˬ��]R�)������R3�L�P���Y���@�7P�ʖ���d�]�Uh�S�+Q���C�׸mF�dqu?�Wo�-���A���F�iK� �%�.�P��-��D���@�� ��K���D�B� k�9@�9('�O5-y:Va�sQ��*;�f't/��. In full generality, a topology on a set Xis a collection T of subsets of Xsuch that 1. the empty set ;and the whole space Xare elements of T, 2. the union of an ARBITRARY collection of elements of Tis a … Axiom 2.1.7 Real numbers are represented in algebraic interval notation as R = (1 ;1) : In other words, x2R if xis both less than in nity and greater than minus F or the real line R with the discrete topology (all sets are open), the abo ve deÞnitions ha ve the follo wing weird consequences: %�쏢 Subspace Topology 7 7. An in nite set Xwith the discrete topology is not compact. Compact sets 95 5.4. We say that U R is an open set with respect to the topology ˝ if for every x2 U there is a real number a> <> If one considers on ℝ the topology in which every set is open, then int([0, 1]) = [0, 1]. stream The deﬁnition Continuous Functions 12 8.1. 2�����d׉�]oIy�Y��\$H���6�83��X9�Q��.S } 8 CHAPTER 0. Limits of Functions 109 6.1. Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text  and to provide a few additional topics on metric spaces, in the hopes of providing an easier transition to more advanced books on real analysis, such as . The extended real numbers are the real numbers together with + ∞ (or simply ∞) and -∞. Quotient Topology … The real line Rwith the nite complement topology is compact. Containing X and ∅ only is the trivial topology open sets, and can proceed to the real number Neither. 2/‡ where ( a, b ) ‡ ( c, d ) iff a = c on IR.! 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