# Thread: If M is disconnected....

1. ## If M is disconnected....

Let (M,d) be a metric space. Prove the equivalence of the following:
(i)M is disconnected
(ii)There exists two nonempty, disjoint closed sets in M whose union is M
(iii)There exists two nonempty, disjoint open sets in M whose union is M
(iv)There exists a set A in M such that emptyset != A != M, and A is both open and closed.

these are basically the definitions of a disconnected set aren't they? I have no idea how to prove these =/

2. Originally Posted by cp05
Let (M,d) be a metric space. Prove the equivalence of the following:
(i)M is disconnected
(ii)There exists two nonempty, disjoint closed sets in M whose union is M
(iii)There exists two nonempty, disjoint open sets in M whose union is M
(iv)There exists a set A in M such that emptyset != A != M, and A is both open and closed.

these are basically the definitions of a disconnected set aren't they? I have no idea how to prove these =/
Well what's your definition of disconnected? I assume that $\displaystyle X$ can be written as the disjoint union of open sets.

$\displaystyle i)\implies ii)$: This is clear since if $\displaystyle A\amalg B=X$ (where $\displaystyle A,B$ are open) then $\displaystyle A=X-B$ and $\displaystyle B=X-A$ both of which are disjoint and closed. So then, $\displaystyle X=A\amalg B=\left(X-B\right)\amalg \left(X-A\right)$

$\displaystyle iii)\implies IV)$ Suppose that $\displaystyle A\amalg B=X$ is a disconnection of $\displaystyle X$ (with $\displaystyle A,B$ closed). Then, $\displaystyle B$ is closed by assumption and $\displaystyle B=X-A$ and since $\displaystyle A$ is closed it folllows that $\displaystyle B$ is open.

Oops...I just noticed that I supposed wrong your definition of disconnected. What is it?

3. Disconnected if there exists sets A and B in M such that
1. A != null != B
2. D= A U B
3. clos(A) n B=null=A n clos(B) iff A n B=null

4. Originally Posted by cp05
Disconnected if there exists sets A and B in M such that
1. A != null != B
2. D= A U B
3. clos(A) n B=null=A n clos(B) iff A n B=null
Oh god, separated sets. That's so gross. I don't understand the last part though.

Usually it is "$\displaystyle X=A\cup B$ where $\displaystyle A,B$ are non-empty and $\displaystyle \overline{A}\cap B=B\cap\overline{A}=\varnothing$"

Is that what you mean to say?

5. yes

6. Originally Posted by cp05
yes
Look here.