I'm learning Complex Analysis, and we are given the following definitions:
Definition. Suppose that $\Omega \subseteq$ C and that $\Omega$ is open.
(1) The set $\Omega$ is connected if any two points of $\Omega$ can be joined by a polygonal path lying inside $\Omega.$
(2) The set
$\Omega$ is simply connected if the interior of every simple closed polygonal
path in $\Omega,$
lies in $\Omega$
that is, if “$\Omega$
has no holes”.
(3) The set is a domain if it is connected as well as open.
Later, my note makes a remark saying:
Note that connected is not defined for closed sets, but there are questions about closed sets being connected.
I don't understand this, it says that it's not defined for closed sets, yet it also says there are questions regarding closed sets being connected..?
Why is do closed sets not have connectedness defined?