the closure, interior and the set of isolated points

[math25]

Hi,
Can someone please help me with this problem. I have hard time figuring out what is the closure, interior and the set of isolated points of the set {n + 1/k} where n and k are natural numbers.

thanks

math 25

 

[pka]

(W)hat is the closure, interior and the set of isolated points of the set {n + 1/k} where n and k are natural numbers.

Let's agree for this question \(\displaystyle \mathbb{N}=\mathbb{Z}^+\).

Let \(\displaystyle \mathcal{J}=\left\{n+\frac{1}{k}:\{n,k\}\subset\mathbb{N}\right\}\).

I going to play the back-of-the-book game with you.
I give you the answers. YOU have to supply the proof (reason).

Closure: \(\displaystyle \overline{\mathcal{J}}=\mathcal{J}\cup \{1\}\)

Interior: \(\displaystyle \mathcal{J}^O=\emptyset\)

Isolated points: \(\displaystyle \mathcal{J}\).

 

[math25]

thanks a lot, I think i got it now.

I just have one more question....Is every dense subset of R countable? (my answer is No?)

thank you so much

 

[pka]

I just have one more question....Is every dense subset of R countable? (my answer is No?)

The irrationals are a dense subset of the reals. SO?

 

[math25]

irrationals are uncountable yeah, that exactly what I thought of first, but just wanted to make sure

 

[math25]

thanks for all your help !