[MOVED] Compactness of a Set: [1,3), the naturals, {1/n}

[meks0899]

I'm not sure how to describe each of these.

Prove that each of the following sets in not compact by describing an open cover for it that has no finite subcover.

A. [1,3)
B. ?
C. {1/n : n ? ?}

Thanks

 

[galactus]

Prove that each of the following sets in not compact by describing an open cover for it that has no finite subcover.

A. [1,3)

Let \(\displaystyle \mathcal{F}=B_{n}, \;\ \text{where} \;\ B_{n}=(0,3-\frac{1}{n}), \text{for} \;\ n\in \mathbb{N}\), Then, \(\displaystyle \mathcal{F}\) is an open cover for [1,3), which has no finite subcover.

B. ?

Let \(\displaystyle \mathcal{F}=B_{n}, \;\ B_{n}=(-n,n)\). Then, \(\displaystyle \mathcal{F}\) is an open cover of \(\displaystyle \mathbb{N}\) which has no finite subcover.

C. {1/n : n ? ?}

Think along the same lines as the other two. Let \(\displaystyle \mathcal{F}=B_{n}\), where \(\displaystyle B_{n}=(\frac{1}{n},2), \;\ \text{for} \;\ n\in \mathbb{N}\)

Then \(\displaystyle \mathcal{F}\) is an open cover of \(\displaystyle \mathbb{N}\) which has no finite subcover.


Thanks[/quote]