Distance between a point and an empty set: meaning and value? On page 253 in General Topology by R Engelking:

The distance $\rho(x, A)$ from a point $x$ to a set $A$ in a metric space $(X,\rho)$ is defined by letting $\rho(x, A) = \text {inf}\ {\{\rho(x, a) : a \in A}\},\ \text {if}\ A \ne \emptyset, \text {and}\ \rho(x, \emptyset) = 1$.

Why $\rho(x, \emptyset) = 1$? I mean:
1- What $\rho(x, \emptyset)$ means?
2- Why it is equal to $1$?
 A: The distance $\rho(x,A) = \inf\{\rho(x,a) \mid a \in A\}$ is usually known as the Hausdorff distance and also generalises nicely to a distance between two sets.
Now, what should $\rho(x,\emptyset)$ be? Intuitively, $\rho(x,A)$ gives the smallest distance between $x$ and any element in $A$. But since there are no elements in $\emptyset$, it makes some sense to let $\rho(x,\emptyset)$ be as big as possible, i.e. $\rho(x,\emptyset) = \infty$. This also corresponds to the practice of defining $\inf \emptyset = \infty$ which is common in other areas of mathematics as well.
When we have any metric $\rho$, we can define the bounded metric corresponding to $\rho$ by
$$\overline{\rho}(x,y) = \min\{\rho(x,y),1\},$$
where $1$ is chosen fairly arbitrarily as the bound, but is an often used value. With such a bounded metric, $1$ is in fact the largest possible value, so in this setting it would make sense to let $\rho(x,\emptyset) = 1$.
So why does the author define $\rho(x,\emptyset) = 1$? Without reading the text, I don't know, but at the end of the day, it can be defined as you like. Maybe the author has in mind some bounded metric, or maybe they are just trying to avoid dealing with infinity.
