Connectedness of the closed interval

In summary, the conversation involves a student studying for a topology exam and trying to prove that the closed interval [0,1] is connected. They have come up with a proof that is different from what they have in their notes and are seeking verification. The proof involves assuming the interval can be separated into clopen, disjoint, and non-empty sets, but ultimately concludes that this is impossible, making the proof sound. The other person agrees that the proof seems reasonable.
  • #1
jojo12345
43
0
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 [tex] [0,1]=A\cup B [/tex] 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, [tex]z=\text{inf}B\in B [/tex]. Note that [tex]z\not =0[/tex] because [tex]A\cap B=\emptyset[/tex].

Now, because [tex]B[/tex] is open in the interval, there is some [tex]\epsilon >0[/tex] such that [tex]b=(z-\epsilon,z+\epsilon)\cap [0,1]\subseteq B[/tex]. [tex]b[/tex] cannot contain [tex]0[/tex], again, because this would violate the disjointness of [tex]A[/tex] and [tex]B[/tex]. Also, [tex]b[/tex] cannot contain any numbers less than [tex]z[/tex] because [tex]z[/tex] is a lower bound on [tex]B[/tex]. However, the only way this last sentence can be true is if [tex]z=0[/tex], 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?
 
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  • #2
Sounds reasonable enough.
 
  • #3
Thank you.
 

Related to Connectedness of the closed interval

1. What is the Connectedness of the Closed Interval?

The Connectedness of the Closed Interval refers to the property of a closed interval on a real number line, where all points on the interval are connected and there are no gaps or breaks.

2. How is the Connectedness of the Closed Interval determined?

The Connectedness of the Closed Interval is determined by the fact that every point on the interval can be connected to any other point on the interval by a continuous path, without leaving the interval.

3. Why is the Connectedness of the Closed Interval important?

The Connectedness of the Closed Interval is important because it allows for the use of important mathematical concepts such as intermediate value theorem and connectedness in topology.

4. What are some examples of connected closed intervals?

Some examples of connected closed intervals include [0,1], [-2,5], and [-π,π]. These intervals are connected because there are no gaps or breaks between any two points on the interval.

5. Can a closed interval be connected and not closed at the same time?

No, a closed interval must be both connected and closed. If an interval is not closed, it means that it does not contain its endpoints, which would result in a gap or break in the interval, making it not connected.

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