Closed sets in Cantor Space that are not Clopen

In summary, a closed set in Cantor Space is a subset of the space that contains all of its limit points. This means that every point in the set is either contained in the set itself or is a limit point of the set. A closed set differs from a clopen set, which is both closed and open. A clopen set contains all of its limit points and has no boundary points. It is not possible for a closed set to also be open in Cantor Space because it contains all of its limit points, while an open set does not. To determine if a set is closed in Cantor Space, one can check if it contains all of its limit points. Closed sets in Cantor Space are important in mathematics, particularly
  • #1
Bacle
662
1
Hi,

Is there a characterization of subsets of the Cantor space C that are closed but not open? As a totally-disconnected set/space, C has a basis of clopen sets; but I'm just curious of what the closed non-open sets are.
 
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  • #2
I guess the subsets which don't contain a 'part' of C ( i.e. a small copy of C) are among such sets.These include the subsets containing finitely many points; which are closed but not open.
 

FAQ: Closed sets in Cantor Space that are not Clopen

1) What is a closed set in Cantor Space?

A closed set in Cantor Space is a subset of the space that contains all of its limit points. In other words, every point in the set is either contained in the set itself, or is a limit point of the set.

2) What is the difference between a closed set and a clopen set in Cantor Space?

A closed set in Cantor Space is a set that contains all of its limit points, while a clopen set is both closed and open. This means that a clopen set contains all of its limit points and has no boundary points.

3) Can a closed set in Cantor Space be open?

No, a closed set in Cantor Space cannot be open because it contains all of its limit points. An open set, on the other hand, does not contain its limit points.

4) How can you determine if a set in Cantor Space is closed?

To determine if a set in Cantor Space is closed, you can check if it contains all of its limit points. If every point in the set is either contained in the set itself or is a limit point of the set, then the set is closed.

5) Are closed sets in Cantor Space important in mathematics?

Yes, closed sets in Cantor Space are important in mathematics because they have many applications in topology, analysis, and other branches of mathematics. They play a crucial role in understanding the structure of a space and its properties.

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