Given the Point set S - Part (c)

[nasi112]

Given the point set [MATH]S: \{i, \frac{i}{2}, \frac{i}{3}, \frac{i}{4}, . . .\}[/MATH]
(c) Is [MATH]S[/MATH] closed?

 

[Deleted member 4993]

Given the point set [MATH]S: \{i, \frac{i}{2}, \frac{i}{3}, \frac{i}{4}, . . .\}[/MATH]
(c) Is [MATH]S[/MATH] closed?

What do you think and

why do you think so?

 

[pka]

Given the point set [MATH]S: \{i, \frac{i}{2}, \frac{i}{3}, \frac{i}{4}, . . .\}[/MATH](c) Is [MATH]S[/MATH] closed?

The statement that the set \(\mathcal{S}=\left\{\dfrac{i}{n}:n\in\mathbb{Z}^+\right\}\) is closed means that \(\mathcal{S}\) contains all of its limit points.
The set \(\mathcal{T}=\left\{{i}\cdot{n}:n\in\mathbb{Z}^+\right\}\) has no limit point so the set is closed.
But because that you do not know what a neighborhood of a point is that is. Thus there no way for you to understand if the set \(\mathcal{S}\) has any limit points.
Is \(\mathcal{S}\) closed or not?