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## Main Question or Discussion Point

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?

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?