7.2 Distance Between Sets
Definition 7.2.1 (Distance Between Sets). Let $X$ be a set, $d: X \times X \to [0, \infty)$ be a pseudometric, and $A, B \subset X$, then
is the distance between $A$ and $B$.
Proposition 7.2.2. Let $(X, d)$ be a pseudometric space, $A \subset X$, and
then $d_{A} \in UC(X; [0, \infty))$, where for any $x, y \in X$, $|d_{A}(x) - d_{A}(y)| \le d(x, y)$.
Proof. Let $x, y \in X$, then
As the argument is symmetric, $|d_{A}(x) - d_{A}(y)| \le d(x, y)$.$\square$
Definition 7.2.3 (Fattening). Let $X$ be a set, $d: X \times X \to [0, \infty)$ be a pseudometric, $A \subset X$, and $\eps > 0$, then
is the $\eps$-fattening of $A$ with respect to $d$.
Proposition 7.2.4. Let $(X, d)$ be a metric space and $A \subset X$, then
Proof. By Proposition 5.1.13.$\square$
Proposition 7.2.5. Let $(X, d)$ be a metric space, $C \subset X$ be closed, and $K \subset X$ be compact. If $C \cap K = \emptyset$, then $d(K, C) > 0$.
Proof. Suppose that $d(K, C) = 0$, then there exists $\seq{(x_n, y_n)}\subset K \times C$ such that $d(x_{n}, y_{n}) \to 0$ as $n \to \infty$. By Definition 4.16.1, there exists a subsequence $\seq{n_k}$ and $x \in K$ such that $x_{n_k}\to x$ as $k \to \infty$. In which case, $d(x, C) = 0$, and $x \in \ol{K}$ by Proposition 7.2.4, so $K \cap C \ne \emptyset$.$\square$