Thread: Prove that the closed interval [0,1] is a closed set

1. Prove that the closed interval [0,1] is a closed set

Prove that the closed interval [0,1] is a closed set and that the open interval (0,1) is an open set..

Could I just say that for [0,1], every open ball B(0,r), r > 0 contains at least one point less than 0 and therefore not an element of [0,1] to prove it is closed

and for (0,1), its complement (-inf, 0] U [1, inf) must be closed by similar reasoning?

is there a way to incorporate the theorem:

a set S is closed iff for all sequences {xk} such that xk is an element of S for each k and xk --> x, it is the case that x is an element of S

I feel like Im basically just stating the definition of closed/open sets up top, and I don't know if that's an acceptable way to do it

Thanks

2. Re: Prove that the closed interval [0,1] is a closed set

Originally Posted by JaredG
Prove that the closed interval [0,1] is a closed set and that the open interval (0,1) is an open set..
Could I just say that for [0,1], every open ball B(0,r), r > 0 contains at least one point less than 0 and therefore not an element of [0,1] to prove it is closed
and for (0,1), its complement (-inf, 0] U [1, inf) must be closed by similar reasoning?
There are as many ways to prove this as there are people reading this.
If $\displaystyle a<0$ then $\displaystyle a\in\left( { - \infty ,\frac{a}{2}} \right)$ and $\displaystyle \left( { - \infty ,\frac{a}{2}} \right)\cap [0,1]=\emptyset$
A simpler statement if $\displaystyle b>1$ proving the complement of $\displaystyle [0,1]$ is open.

3. Re: Prove that the closed interval [0,1] is a closed set

Thank you for responding but I cant make sense of that..

I was asking if the method I was suggesting would be sufficient.

4. Re: Prove that the closed interval [0,1] is a closed set

Originally Posted by JaredG
Thank you for responding but I cant make sense of that..
I was asking if the method I was suggesting would be sufficient.
I guess that make us even because I can't make sense of what you posted.
But then I have only taught this material for forty years.

Do you know how to prove sets of real numbers are open or closed?

5. Re: Prove that the closed interval [0,1] is a closed set

Sorry I'll try to write it more formally.

Using this definition of an open set: A set S in Rn is open if for every element x in S, there is an r > 0 such that B(x,r) is a subset of S

suppose [0,1] is open

[0,1] contains the element 0

B(0,r) contains (0 - r) which is not an element of [0,1] for any r > 0

therefore B(x,r) is not a subset of S for all x in S, which implies [0,1] must be closed by contradiction

Do you know how to prove sets of real numbers are open or closed?
No, I thought that's what was being asked here though?

Im interested in the proofs you suggested, could you explain them to someone who hasn't taught the material for 40 yrs? I don't understand how that intersection being empty says anything about [0,1] being closed. I could replace [0,1] with (0,1) and the statement would still be true..

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(0,1) is closed for

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