Thread: a strange question

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

since we already know E 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?

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 AUB, prove that either E A or E B.
since we already know E 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 then if then .

That is why we require to be connected.

Originally Posted by Plato

Let then if then .

That is why we require to be connected.

hmm ic, so how can I use the connectedness to prove this?

Originally Posted by wopashui

so how can I use the connectedness to prove this?

Let .
If both of those are nonempty, then they are separated.
But .
What is wrong with that?

Originally Posted by Plato

Let .
If both of those are nonempty, then they are separated.
But .
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?

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?

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?

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 are separated means that each is non-empty and neither contains a point nor a limit point of the other.

separated means disjoint, right?

Originally Posted by wopashui

separated means disjoint, right?

Well it is more that just that, although they are disjoint.

Originally Posted by wopashui

separated means disjoint, right?

their closure is disjoint from the other set