# Thread: Metric spaces distances... Help

1. ## Metric spaces distances... Help

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

2. Originally Posted by Dreamer78692
True or False.. If true prove it or else give a counter example
(g) A ⊂ B ⇒ dist (B;C) ≤ dist (A;C)
True. I can visualize it, but cant proof this 1 formally
If $\displaystyle N\subseteq M\subseteq \mathbb{R}$ then is it true $\displaystyle \inf(M)\le\inf(N)~?$

If $\displaystyle x\in C~\&~a\in A$ is it true that is it true that $\displaystyle d(a,x)\in \{d(y,x):y\in B\}~?$

3. So since the inf(B)≤ inf(A)
then dist (B;C) ≤ dist (A;C)

From $\displaystyle \{d(x,y):x\in A~\&~y\in C\}\subseteq\{d(z,w):z\in B~\&~w\in C\}$ that follows.