Easy question about a closed set.

Question

Please be lenient. I understand that this question depends on my misunderstanding but I am a newbie.

$X=\{A,B,C\}$ is a set. We define the topology on $X$ as $\tau =\mathscr{P}(X).$ As I know it is the discret topology. By definition all elements in this topology is open subsets.

Okey. Now I want to look at the closed subset. By definition the complement of a open subset is a closed subset.

Let's take for example the folowing open subset $M={A}$. It is the element of our topology. $X\setminus M=\{B,C\}.$ It must be closed. But it is also is the element of our topology which must be open.

Where am I wrong? What do I misunderstand?

Answer 1

You are right. All sets are also closed in the discrete topology.

Answer 2

As Munkres nicely writes in his book: "topology is not like doors, where a door is either open or closed: an open set may be closed as well". Hence, it is perfectly fine for a set that to be both open and closed. And indeed this is the case for any discrete topology.