ConsiderthesetR1ofallreal-valuedfunctionsontheunitinterval.
1.Foreach f ?R1,eachfinitesubsetFof...

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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.

,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

David · Accepted Answer

1
1. Let U(f, F1, δ1) and U(f, F2, δ2) be given. Let U(f, F, δ) where F = F1 ∪ F2 and δ

ConsiderthesetR1ofallreal-valuedfunctionsontheunitinterval. 1.Foreach f ?R1,eachfinitesubset F of Iandeachpositive d , let U(f,F , d)={ g ?R1|g(x) -f(x)| ,foreach X ?F} ShowthatthesetsU(f,F,...

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