Can $ \ \ \sup_{x \in X} \inf_{y \in Y} \ d(x,y) \neq 0$?

Question

Let $X, \ Y$ be two non-empty subset of a metric space $ (M,d)$ such that $X \subset Y$.

My question is-

Can $ \ \ \sup_{x \in X} \inf_{y \in Y} \ d(x,y) \neq 0$ ?

My calculation shows that always $ \ \ \sup_{x \in X} \inf_{y \in Y} \ d(x,y) = 0$

score 2 · Accepted Answer · answered Jul 23 '19 at 11:31

2

$inf_{y\in Y}d(x,y)=0$ since $x\in Y$ therefore the sup is always zero.

answered Jul 23 '19 at 11:31

Tsemo Aristide

But see this example-https://math.stackexchange.com/questions/2689787/seeking-an-example-where-the-hausdorff-distance-between-two-non-empty-closed-sub – MAS Jul 23 '19 at 11:32
@M.A.SARKAR "But see this example": That example is irrelevant, because $\sup_{x\in X}\inf_{y\in Y}d(x,y)$ is not $d_H(X,Y)$. – David C. Ullrich Jul 23 '19 at 16:32
@DavidC.Ullrich, yes you are right – MAS Jul 23 '19 at 16:34

1 Answers1