What are the origins of the ubiquitous "distance between subspaces" in a metric space?

What are the origins of the ubiquitous "distance between subspaces" in a
metric space?

Let $(X,d)$ be a metric space with subspaces $A, B \subset X$. The
following notion of "distance" or "separation" is natural and shows up
absolutely everywhere:
$$\delta_X(A,B) = \inf_{a \in A, b \in B}d(a,b)$$
Here's my question:
When did this $\delta_X$ first appear in the literature?
More particularly, I'm not interested in specializations to folklore like
the distance between a point and a line in Euclidean space, but rather of
$\delta$ as an interesting object in the general theory of metric spaces.
This question is a reference request only. I'd just like to properly
attribute this definition to its originator(s) -- if they can be found --
whenever I use it.