Thread: 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 =/

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 can be written as the disjoint union of open sets.

: This is clear since if (where are open) then and both of which are disjoint and closed. So then,

Suppose that is a disconnection of (with closed). Then, is closed by assumption and and since is closed it folllows that is open.



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

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

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 " where are non-empty and "

Is that what you mean to say?

yes

Originally Posted by cp05

yes

Look here.