Let $X$ be a metric space and $K$(Compact), $C$(Closed) $⊂X$ such that $K∩C=∅.$Show that $d(K,C)>0.$

Question

Please tell if my proof works for the following problem:

Let $X$ be a metric space and $K$(Compact), $C$(Closed) $⊂X$ such that $K∩C=∅.$Show that $d(K,C)>0.$

SOLUTION: $d_C:X\to\mathbb R$ is continuous $\implies d_C:K\to\mathbb R$ is continuous where $K$ is compact$\implies\exists~k\in K$ such that $$d_C(k)=\inf_{~x~\in K}d_C(x)=\inf_{x\in K} d_C(x)=\inf_{x~\in K}\inf_{c~\in C}d(x,c)=\inf_{x\in K,~c~\in C}d(x,c)=d(K, C)$$If possible let $d(K,C)=0\implies d_C(k)=0\implies k\in\overline C=C!\\\text{Hence the result follows}.$

Related : https://math.stackexchange.com/questions/48714/a-and-b-disjoint-a-compact-and-b-closed-implies-there-is-positive-distance-bet — Arnaud D., Nov 28 '17 at 10:04

score 0 · Accepted Answer · answered Nov 07 '13 at 08:16

0

Suppose $\inf_{c \in C, k \in K} d(c,k) = 0$. Then there exist $c_n \in C, k_n \in K$ such that $d(c_n,k_n) < \frac{1}{n}$. Since $K$ is compact we can presume that $k_n \to k \in K$.

Since $d(c_n,k) \le d(c_n,k_n)+d(k_n,k)$, we see that $d(c_n,k) \to 0$, that is $c_n \to k$. Since $C$ is closed, we have $k \in C$, a contradiction.

Hence $\inf_{c \in C, k \in K} d(c,k) > 0$.

answered Nov 07 '13 at 08:16

copper.hat

172,524

Where did my proof go fail to work? – Shantipriya Nov 07 '13 at 08:16
You need to show that $d_C$ is continuous and you need to show that $d_C(k) = 0$ implies $k \in C$. – copper.hat Nov 07 '13 at 08:19
Actually I considered the results as standard. Is it correct otherwise? – Shantipriya Nov 07 '13 at 08:21
It depends on what you are looking for. The result itself is standard and well known, so presumably this is for the learning experience. Continuity of $d_C$ and the fact that $d_C(x) = 0$ iff $x \in C$ are the essence of the proof, so if you assume them, then there isn't much left to prove? – copper.hat Nov 07 '13 at 08:25
Generally I would avoid introducing $d_C$ unless it serves a later purpose. But this is a style issue and very subjective. – copper.hat Nov 07 '13 at 08:26
Thank you. Indeed your proof is elegant. – Shantipriya Nov 07 '13 at 08:28

Let $X$ be a metric space and $K$(Compact), $C$(Closed) $⊂X$ such that $K∩C=∅.$Show that $d(K,C)>0.$

1 Answers1