Question

(b) Find an example of an open set G in a metric space X and a closed subset F of G such that there is no δ > 0 with {x : dist(x, F) < δ} C G

Answer 1

To find an Example of an open set G in a metric space X and a closed subset F of G Let X = R with Usual metric F = [-infinity , 0] let G = (-infinity , E/q) is open set where E > 0 & E < delta also , F Subset G In a real line , every closed and bounded set is compact and Vice - versa Therefore , F is closed but not bounded I e F also not compact Then for each delta > 0 E elementof {x: dist [x , F] < delto) but elementof notelement (-infinity , E/2 \r\n Therefore , (2) [x: dist (x , F) < delta] notelement G for each delta > 0 Thus , There is no delta > 0 with {x: dist (x , F) < delta} < G