TOPOLOGY OF SPACES 1 answer below »


ConsiderthesetR1ofallreal-valuedfunctionsontheunitinterval.

1.Foreach f ?R1,eachfinitesubsetFof Iandeachpositive d,let

U(f,F, d)={g?R1|g(x) -f(x)| ,foreachX?F}

ShowthatthesetsU(f,F, d)formaneighborhood baseat f,making R1atopologicalspace.

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2.Foreachf ?R1 theclosureoftheone-point set{f}isjust{ f}. (Thisisnotunusual.Infact,itisasituationtobedesired;spaceswithoutthispropertyaredifficulttodealwith)

3. Forf ?R1ande>0, let

V(f,e)={g?R1lg(x) -f(x)lx?I}

VerifythatthesetsV(f,e)formaneighborhood baseat f,makingR1atopologicalspace.

4.Comparethetopologiesdefinedin1and3

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