Identifying Compact Sets in the Slitted and Moore Planes: What's the Method?

[ForMyThunder]

Is there any easy way to find all the compact sets of 1) the slitted plane and 2) the Moore plane?

1) defined as the topology generated by a base consisting of z\cup A where A is a disc about z with finitely many lines deleted. I believe the compact sets in this topology coincide with the usual real line. How do I prove this?

2) (rough sketch) I know that a closed, bounded set disjoint from the x-axis is compact. I guess if a set touched the x-axis in an infinite number of points, it would not be compact. So the compact sets consisting of a closed, bounded set disjoint from the x-axis along with finitely many points on the x-axis is compact?

 

[fresh_42]

1) I do not think so. A closed section of disc should be compact and is no real line.
2) The number of points on the $x-$ axis shouldn't play a role here.