Compact Sets of Moore Plane

  • #1
147
0

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?
 

Answers and Replies

  • #2
13,221
10,138
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.
 

Related Threads on Compact Sets of Moore Plane

Replies
3
Views
3K
  • Last Post
Replies
12
Views
1K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
13
Views
2K
Replies
1
Views
897
Replies
6
Views
994
Replies
7
Views
5K
Replies
6
Views
3K
  • Last Post
Replies
8
Views
2K
Replies
2
Views
1K
Top