Proving A is Contained in M or N of Disconnected X

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In summary, if A is a connected subset of a disconnected set X with non-empty closed disjoint sets M and N, then A must be contained in either M or N. This can be proven by assuming that A intersects both M and N, and showing that this leads to a contradiction. Therefore, either AnM or AnN must be empty, and A is contained in either M or N.
  • #1
math8
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If A is a connected subset of a disconnected set X s.t. X=MUN , M,N nonempty closed disjoint sets, how do we show, A is either contained in M or in N?

I can start a proof, but then, I am kind of stuck.
I would go by contradiction and say A intersection M is non empty and A intersection N is non empty. Hence A would be the union of 2 non empty disjoint sets. But since A is connected, A intersection M and A intersection N cannot be both open. So without loss of generality, say A intersection M is not open. Hence X\(A intersection M) is not closed.
Then I get stuck. Any help?
 
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  • #2
X is disconnected. That means, there are at least two nonempty disjoint open-and-closed sets. You are conveniently given these two sets. They are M and N.

If A is connected, that means that, relative to the topology on X, there is only one non-empty disjoint open-and-closed set. Namely, A itself.

Now, suppose that A intersects both M and N. What can you say about AnM and AnN? For both, is the set empty? Is it closed?
 
  • #3
I guess AnM and AnN are both clopen. But since A is connected, one of the AnM or AnN has to be empty. Say, AnM is empty. So ACN.

Is that right?
 
  • #4
assume that AnM and AnN are non empty and A is connected. Since M and N are a disconnection of X they are open in X and by the subspace topolgoy A'=AnM and A''=AnN are open and disjoint in A and form a disconnection of A. contradiction QED
 

1. What does it mean for A to be contained in M or N of Disconnected X?

When we say A is contained in M or N of Disconnected X, it means that A is a subset of either M or N, where M and N are disjoint sets. This means that A contains elements that are only found in M or only found in N, but not in both sets.

2. How can we prove that A is contained in M or N of Disconnected X?

To prove that A is contained in M or N of Disconnected X, we need to show that all elements of A are either in M or N, and that there are no elements in A that are not in M or N. This can be done through logical reasoning, mathematical proofs, or empirical evidence.

3. What is the significance of proving A is contained in M or N of Disconnected X?

Proving that A is contained in M or N of Disconnected X can help us better understand the relationship between different sets and their subsets. It can also be useful in solving problems and making predictions in various fields such as mathematics, physics, and computer science.

4. Can A be contained in both M and N in Disconnected X?

No, in Disconnected X, M and N are disjoint sets, meaning that they have no elements in common. Therefore, A can only be contained in either M or N, but not both.

5. How does proving A is contained in M or N of Disconnected X relate to the concept of disjoint sets?

Proving that A is contained in M or N of Disconnected X is closely related to the concept of disjoint sets. Disjoint sets are sets that have no elements in common, which means that if A is contained in either M or N, it cannot have any elements in common with the other set. This further strengthens the proof that A is contained in M or N.

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