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,...

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