# Give an example of a set which has exactly 3 limit points and the set is closed.

So the question is,

Give an example of a set of real numbers which has exactly 3 limit points and the set is closed.

I have been trying to solve it by myself, and I know one potential set can be $$A$$, such that, $$A = \{k + \frac{1}{n}\ : n \in \mathbb{N} \}$$ Where $$K$$ is the limit point of $$A$$.

The problem with this set is, this set is not closed, by the definition of a closed set, the set also need to contain it's limit points, but $$k \notin A$$

• Hint: Look at the closure of that set, which just adds $k$. Now do it 3 times
– Alan
Jul 2 at 10:44
• What do you mean by closure of a set? Sorry I am very new in real analysis... Jul 2 at 10:47
• The closure of a set is the smallest closed set it is a subset of, you get it by adding all the limit points
– Alan
Jul 2 at 10:50

Let $$A(k)=\{k+\frac{1}{n}\mid n\in\mathbb{N}\}\cup \{k\}$$, then $$A(k)$$ is closed and has one limit point, ie $$k$$. Now take $$A(0)\cup A(1) \cup A(2)$$. which is a finite union of closed sets, and hence closed. It contains three limit points, ie 0, 1, and 2, and it's easy to check it doesn't contain any others.