7. Let Ω be open, E ⊆ Ω be a set with no limit points in Ω, and C ⊆ Ω \ E be a simple closed contour such that the inside of C lies in Ω. (The set E could be the set of isolated singularities of an...

7. Let Ω be open,
E

⊆
Ω be a set with no limit points in Ω, and
C

⊆
Ω
\

E
be a simple closed contour such that the inside of
C
lies in Ω. (The set
E
could be the set of isolated

singularities of an analytic function.) Why can only finitely many points of
E
lie inside of
C?