Could someone please check my answers, especially if my proves are enough to answer the question.
True or False.. If true prove it or else give a counter example
(a) dist (x; {x}) = 0.
True. Since the inf(x,{x})=0.
(b) x ∈ A ⇒ dist (x;A) = 0.
True. Since x ∈ A, the inf(x,y|y∈A)=0.
(c) dist (x;A) = 0 ⇒ x ∈ A.
False. eg. x=5 ; A=(5,6]
(d) For any subsets A; B; C of a metric space X,
dist (A;B) ≤ dist (A;C) + dist (C;B).
False. eg. A=[0,2], B=[4,6], C=[1,5]
(e) A ∩ B ̸= ∅ ⇒ dist (A;B) = 0
True. Since there exists an x ∈ A and x ∈ B then the inf(x,x)=0
(f) dist (A;B) = 0 ⇒ A ∩ B ̸= ∅
False. A=[0,2] B=(2,3)
(g) A ⊂ B ⇒ dist (B;C) ≤ dist (A;C)
True. I can visualize it, but cant proof this 1 formally