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General TopologyA Solution Manual forWillard[2004]Jianfei ShenSchool of Economics, The University of New South WalesSydney, AustraliaOctober 15, 2011
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See morePrefaceSydney,Jianfei ShenOctober 15, 2011v
ContentsPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v1Set Theory and Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32Topological Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.1Fundamental Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.2Neighborhoods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132.3Bases and Subbases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163New Spaces from Old. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193.1Subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193.2Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193.3Product Spaces, Weak Topologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243.4Quotient Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .314.1Inadequacy of Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .314.2Nets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324.3Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345Separation and Countability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375.1The Separation Axioms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375.2Regularity and Complete Regularity. . . . . . . . . . . . . . . . . . . . . . . . . . . .395.3Normal Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .405.4Countability Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .416Compactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .436.1Compact Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45ix
AcronymsRthe set of real numbersIOE0; 1ŁPRXQxi
1SET THEORY AND METRIC SPACES1.1Set Theory1A. Russell’s ParadoxIExercise1.The phenomenon to be presented here was first exhibited byRussell in 1901, and consequently is known asRussell’s Paradox.Suppose we allow as sets thingsAfor whichA2A. LetPbe the set of allsets. ThenPcan be divided into two nonempty subsets,P1D˚A2PWA…AandP2D fA2PWA2Ag. Show that this results in the contradiction:P12P1[]P1…P1. Does our [naive] restriction on sets given in 1.1 eliminate thecontradiction?Proof.IfP12P1, thenP12P2, i.e.,P1…P1. But ifP1…P1, thenP12P1. Acontradiction.ut1B. De Morgan’s laws and the distributive lawsIExercise2.a.AXT2BDS2.AXB /.b.B[T2BDT2.B[B /.c.IfAnmis a subset ofAfornD1; 2; : : :andmD1; 2; : : :, is it necessarily truethat1[nD1241\mD1Anm35D1\mD1241[nD1Anm35‹Proof.[a]Ifx2AXT2B, thenx2Aandx…T2B; thus,x2Aandx…Bfor some, sox2.AXB /for some; hencex2S2.AXB /.On the other hand, ifx2S2.AXB /, thenx2AXBfor some2,i.e.,x2Aandx…Bfor some2. Thus,x2Aandx…T2B; that is,x2AXT2B.1
2CHAPTER 1SET THEORY AND METRIC SPACES[b]Ifx2B[T2B, thenx2Bfor all, thenx2.B[B /for all, i.e.,x2T2.B[B /. On the other hand, ifx2T2.B[B /, thenx2.B[B /for all, i.e.,x2Borx2Bfor all; that is,x2B[T2B.[c]They are one and the same set.ut1C. Ordered pairsIExercise3.Show that, if.x1; x2/is defined to be˚fx1g;fx1; x2g, then.x1; x2/D.y1; y2/iffx1Dy1andx2Dy2.Proof.Ifx1Dy1andx2Dy2, then, clearly,.x1; x2/D˚fx1g;fx1; x2gD˚fy1g;fy1; y2gD.y1; y2/. Now assume that˚fx1g;fx1; x2gD˚fy1g;fy1; y2g.Ifx1¤x2, thenfx1g D fy1gandfx1; x2g D fy1; y2g. So, first,x1Dy1and thenfx1; x2g D fy1; y2gimplies thatx2Dy2. Ifx1Dx2, then˚fx1g;fx1; x1gD˚fx1g.
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Topology, Metric space, Proof Let, proof, X