What is the boundary of an open subset $\Omega \subset \Bbb H$ 
Let $\Bbb H:=\{(x,y): x\in \Bbb R, y>0\}$ and $\Omega$ an open subset of $\Bbb H$. Then what is $\partial \Omega$ ?

Of course $\partial \Omega =\overline{\Omega} \setminus {\Omega}$. Now the point is while considering $\overline{\Omega}$ is it $\overline{\Omega}$ as a subset of $\Bbb R^2$ or $\overline{\Omega} \cap \Bbb H$ ? For instance, let $\Omega=\{(x,y) :x \in \Bbb R, y>0, x^2+y^2<1\}$ . Then $\Omega$ is clearly an open subset of $\Bbb H$, then is $\partial \Omega = \{(x,y) :x \in \Bbb R, y>0, x^2+y^2=1\}$ or $\{(x,y) :x \in \Bbb R, y\ge 0, x^2+y^2=1\} \cup \{(x,0):-1 \le x \le 1\}$ ?

As we are considering it as a subset of the upper half plane, while considering its boundary shouldn't we only consider its boundary in the upper half plane, as we are treating it as the ambient space to start off with?


For your $\Omega$ we have $\Omega=\{(x,y)\in\mathbb H : x^2+y^2<1
\}$ and has
$$\partial\Omega=\{(x,y)\in\mathbb H : x^2+y^2=1
\}.$$
For the closure, this is
$\overline \Omega=\{(x,y)\in\mathbb H : x^2+y^2\le1
\}$
