What is the closure of A when A is not connected?

Question

I have already what is a closure. I saw some examples of closure what A is something kind of this $(0,1)$. But when A is, for example, $(0,1) \cup (2,3)$, What is the closure of this? Is it $[0,1]\cup [2,3]$? So is this closure "closed"?

Answer 1

HINT:

You can find closure with the help of Adherent points definition ,

$\bar{A}=A\cup D(A)$

where , $\bar{A}$ is closure of $A$ and $D(A)$ is the set of all limit points of $A$.

Answer 2

A set is (topologically) closed if it contains all of its limit point in its topological space.   The closure of a set is the smallest closed set containing the set.   Which entails that a closure for a set is the union of the set and its limit points.

The limit points for $(0;1)\cup(2;3)$ are $\{0,1,2,3\}$.   Thus $[0;1]\cup[2;3]$ is the closure, and of course it is closed.

The "hole" between the subintervals is no impediment to closure.   You do not need to fill it in.