ConsiderthesetR1ofallreal-valuedfunctionsontheunitinterval.1.Foreach f ?R1eachfi

ConsiderthesetR1ofallreal-valuedfunctionsontheunitinterval.1.Foreach f ?R1eachfinitesubsetFof Iandeachpositive dletU(fF d)={g?R1|g(x) -f(x)| foreachX?F}ShowthatthesetsU(fF d)formaneighborhood baseat fmaking R1atopologicalspace.2.Foreachf ?R1 theclosureoftheone-point set{f}isjust{ f}. (Thisisnotunusual.Infactitisasituationtobedesired;spaceswithoutthispropertyaredifficulttodealwith)3. Forf ?R1ande>0 letV(fe)={g?R1lg(x) -f(x)lx?I}VerifythatthesetsV(fe)formaneighborhood baseat fmakingR1atopologicalspace.4.Comparethetopologiesdefinedin1and3
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