Connectedness of the closed interval

[jojo12345]

Hi,

I'm studying for the final exam in my first course in topology. I'm currently recalling as many theorems as I can and trying to prove them without referring to a text or notes. I think I have a proof that the closed interval [0,1] is connected, but it's different than what I have in my notes. I was hoping I might just verify that my proof is sound.

Proof:
Assume $$[0,1]=A\cup B$$ where A and B are clopen, disjoint, and not empty. Further, assume that A contains 0. Because B is closed in the interval and the interval is closed in the reals, B is closed in the reals and contains its infimum, $$z=\text{inf}B\in B$$. Note that $$z\not =0$$ because $$A\cap B=\emptyset$$.

Now, because $$B$$ is open in the interval, there is some $$\epsilon >0$$ such that $$b=(z-\epsilon,z+\epsilon)\cap [0,1]\subseteq B$$. $$b$$ cannot contain $$0$$, again, because this would violate the disjointness of $$A$$ and $$B$$. Also, $$b$$ cannot contain any numbers less than $$z$$ because $$z$$ is a lower bound on $$B$$. However, the only way this last sentence can be true is if $$z=0$$, which is absurd. Thus the initial separation of the interval into clopen, disjoint, not empty sets is impossible.

Does this work out, or did I overlook something?

 

[Hurkyl]

Sounds reasonable enough.

 

[jojo12345]

Thank you.