Let X be a non-empty set and let the metric d be defined by d(x, x)...

Transcribed Image Text:

Let X be a non-empty set and let the metric d be defined by d(x, x) = 0 for all x = X and d(x, y) = 1 for all x + y in X. (a) Let 0 < <1 and let x = X. Describe the closure Cl(B.(x)) and the boundary AB(x) of the open ball B(x). [3 marks] (b) Let x = X. Assume that X\ {x} 0. Find all values of (0,1] so that: (i) Cl(B(x)) = {y X : d(x, y) }, [3 marks] (ii) B(x) = {y X : d(x, y) = }. [3 marks] Let X be a non-empty set and let the metric d be defined by d(x, x) = 0 for all x = X and d(x, y) = 1 for all x + y in X. (a) Let 0 < <1 and let x = X. Describe the closure Cl(B.(x)) and the boundary AB(x) of the open ball B(x). [3 marks] (b) Let x = X. Assume that X\ {x} 0. Find all values of (0,1] so that: (i) Cl(B(x)) = {y X : d(x, y) }, [3 marks] (ii) B(x) = {y X : d(x, y) = }. [3 marks]