# Metric between bounded and closed sets?

Define the set:

$$X:=\{K \subset \mathbb{C}: K \text{ is bounded and closed } \}.$$

Define a function $$d : X \times X \rightarrow \mathbb{R}$$ via:

$$d(K_1,K_2) = \inf \{ \delta > 0 : K_1 \subset N_{\delta}(K_2) \text{ and } K_2 \subset N_{\delta}(K_1)\},$$

where

$$N_δ(K) = \{ x \in \mathbb{C} : \exists y \in K \text{ with } |x−y|< \delta \}.$$

Note that we could also write $$N_{\delta}(K) = \bigcup \limits_{y \in K} N_{\delta}(y)$$, where $$N_{\delta}(y) = \{x \in \mathbb{C} : |x-y|<\delta\}$$.

I'm having trouble with understanding $$d$$ and $$N_δ(K)$$. Can someone explain me why $$d(K_1,K_2)$$ is a metric?

• I have edited your post (it is currently being peer reviewed); please check that the edits correspond with your original intent. – Sambo Oct 13 '18 at 18:03

Intuitively $$N_{\delta}$$ takes a set $$K$$ and 'fattens' it by radius of $$\delta$$. An alternative way of writing it would be $$N_\delta(K) = K + B_{\delta}(0)$$ $$d$$ then takes two sets $$K_1$$ and $$K_2$$, and measures how much we need to 'fatten' $$K_1$$ until it contains all of $$K_2$$ and vice versa. Another way of writing it is $$d(K_1, K_2) = \inf \{\delta > 0 : (\forall x \in K_1)(\exists y \in K_2)\ \lvert x - y \rvert \leq \delta \ \text{and}\ (\forall y \in K_2)(\exists x \in K_1)\ \lvert x - y \rvert < \delta\}$$ $$d$$ is well defined since the sets $$K_1$$ and $$K_2$$ are bounded, thus we aren't taking the infinum over an empty set.
By $$d$$'s very definition, it is symmetric and positive. What remains is to show $$d(K_1, K_2) = 0 \iff K_1 = K_2$$ (this requires the compactness of $$K_1$$ and $$K_2$$) and the triangle inequality.
• It's worth nothing that the sets $K_1$ and $K_2$ should also be non-empty for $d$ to be well-defined! – Sambo Oct 13 '18 at 18:15
• @bitesizebo if $d(A,B) = 0$, then every point of A is within zero distance of B,and hence must belong to B since B is compact, so $A \subset B$ and similarly $B \subset A$. So $d(A, B) = 0$ implies that $A = B$. Is this it for definiteness? And how do I go with the triangle inequality? – dxdydz Oct 14 '18 at 14:30