Meaning of $\mathbb K\in\{\mathbb R,\mathbb C\}$? What is the meaning of $\mathbb K\in\{\mathbb R,\mathbb C\}$ in the following?

Let $A$ be a non-empty open subset of $\mathbb K\in\{\mathbb R,\mathbb C\}$ and let $f:A\rightarrow \mathbb K$ be a continuous function.

Is $\mathbb K$ a set of real numbers and the complex numbers?
 A: No, $\Bbb K$ is either the set of real numbers or the set of complex numbers. That's not a way I've seen it written before, but it's straightforward enough: $\{\Bbb R,\Bbb C\}$ is a set that contains the set of real numbers and the set of complex numbers as elements. $\Bbb K$ is an element of that set. So $\Bbb K$ is one of them.
It's the same as saying $n\in \{0,1\}$: $n$ is either $0$ or $1$.
A: It means that $\Bbb K$ is $\Bbb R$ or $\Bbb C$.
A: The author doesn't want to restrict him/herself to only real-valued or complex-valued functions. To leave it open (to define or prove something regardless of this difference) he denotes the field by $\mathbb{K}$ which by this notation can be the reals or the complex numbers (but not another field like the rationals, or a finite field).
It's convenient not to have to say "real- or complex-valued function", but say $\mathbb{K}$-valued function, e.g. 
(IIRC Conway in his functional analysis book uses $\mathbb{F}$ for this, as in "$F$" for field, instead of the classical German abbreviation "$K$" for "Körper", the bold font is mimicking the one for reals and complex numbers ).
A: The author does not want to restrict himself to state a theorem for only the reals $\Bbb R$ or only the complex numbers $\Bbb C$. So instead of writing it down twice he states that it holds for both in the neat short form that it holds for a field $\Bbb K\in \{\Bbb R,\Bbb C\}$, hence for both separately.
