1. ## a strange question

If E is a connected subset of M, and if A and B are disjoint open sets in M with E $\subset$AUB, prove that either E $\subset$A or E $\subset$ B.

since we already know E $\subset$AUB, and A and B are disjoint, so E must be a subset of A or B, why do we even need the connectedness condition here?

2. ## Re: a strange question

Originally Posted by wopashui
If E is a connected subset of M, and if A and B are disjoint open sets in M with E $\subset$AUB, prove that either E $\subset$A or E $\subset$ B.
since we already know E $\subset$AUB, and A and B are disjoint, so E must be a subset of A or B, why do we even need the connectedness condition here?
Let $M=[-4,4],~E=[-2,-1]\cup[1,2],$ then if $A=(-3,0)~\&~B=(0,3)$ then $E\subset A\cup B$.

That is why we require $E$ to be connected.

3. ## Re: a strange question

Originally Posted by Plato
Let $M=[-4,4],~E=[-2,-1]\cup[1,2],$ then if $A=(-3,0)~\&~B=(0,3)$ then $E\subset A\cup B$.

That is why we require $E$ to be connected.
hmm ic, so how can I use the connectedness to prove this?

4. ## Re: a strange question

Originally Posted by wopashui
so how can I use the connectedness to prove this?
Let $E_A=E\cap A~\&~E_B=E\cap B$.
If both of those are nonempty, then they are separated.
But $E=E_A\cup E_B$.
What is wrong with that?

5. ## Re: a strange question

Originally Posted by Plato
Let $E_A=E\cap A~\&~E_B=E\cap B$.
If both of those are nonempty, then they are separated.
But $E=E_A\cup E_B$.
What is wrong with that?
are you showing by contridiction? it seems like it would contradicts that A,B are disjoint, but what is separated mean?

6. ## Re: a strange question

Originally Posted by wopashui
are you showing by contridiction? it seems like it would contradicts that A,B are disjoint, but what is separated mean?
Do you know what it means to say a set is connected?

7. ## Re: a strange question

Originally Posted by Plato
Do you know what it means to say a set is connected?
we just defined eonnected set as not disconnected, if we prove by contridiction, what should we suoopse, what is the neglation?

8. ## Re: a strange question

Originally Posted by wopashui
we just defined eonnected set as not disconnected, if we prove by contridiction, what should we suoopse, what is the neglation?
Connected set as not disconnected.
That is correct. BUT being disconnected means not being the union of two separated sets.

The statement that $A~\&~B$ are separated means that each is non-empty and neither contains a point nor a limit point of the other.

9. ## Re: a strange question

separated means disjoint, right?

10. ## Re: a strange question

Originally Posted by wopashui
separated means disjoint, right?
Well it is more that just that, although they are disjoint.

11. ## Re: a strange question

Originally Posted by wopashui
separated means disjoint, right?
their closure is disjoint from the other set