How to check thet the set is closed but not clopen

It is clear how to proof that the set $$A$$ is open. I just need to find some sequence which elements belong to $$A$$ while its limit does not.

It is also clear that in order to show that the set $$A$$ is closed I need to verify that its complement $$\overline{A}$$ is open.

But sometimes while the set $$A$$ complement $$\overline{A}$$ is open the complement $$\overline{\overline{A}}$$ of this complement $$\overline{A}$$ is also open. So we get clopen set.

I want to show that the set $$A$$ is closed but not clopen. Is it enough to show that $$\overline{A}$$ is open and $$\overline{\overline{A}}$$ is closed? And why it is not follows immidiately that $$\overline{\overline{A}}$$ is closed if I had previously shown that $$A$$ is closed?

Maybe it exist some very simple while sophisticated example of clopen set? It is not clear to me how the set $$\overline{\overline{A}}$$ could be open while the set $$A$$ is closed while them should be the same sets.

Will be very greatfull for help!

• I'm confused with your notation.Do you mean $\overline{A} = X \setminus A$ ? where $X$ is a whole space – Chinnapparaj R Sep 22 '18 at 9:40
• Yes, that is right – Bogdan Sep 22 '18 at 9:56
• Actually $X \setminus (X \setminus A)=A$. So what do you want? you need counterexamples of clopen sets? – Chinnapparaj R Sep 22 '18 at 10:04
• I just can't understand how the set could be clopen and how to show that the set is closed but not clopen. – Bogdan Sep 22 '18 at 10:20
• Which set, specifically, in which topology? – Berci Sep 22 '18 at 10:28

Instead, to prove a set $A$ is open, we need to find a basic open set $U$ (given by a 'basis' for the topology) for each $a\in A$ such that $a\in U\subseteq A$.
The basic open sets in a Euclidean space are the open balls $B_r(x)=\{v:\|v-x\|<r\}$.
In particular in $\Bbb R$ these are the open intervals, and for instance, $\Bbb Q$ is neither open nor closed.
When a nontrivial clopen set $A$ exists in a top.space $X$, the space is called disconnected (we have 2 complementary, nonempty open sets, $A$ and $\bar A=X\setminus A$).
For example, consider $X:=(-1,1)\setminus\{0\}$ with the usual topology (inherited from $\Bbb R$), it's the union of 2 nonempty open sets $(-1,0)$ and $(0,1)$, hence both are clopen in $X$.