some definitions related to open sets

We shall introduce some definitions related to open sets. Here is a metric space $\Omega$  and $A,B\subset\Omega$ .

At first, the closed set means the complementary set of a open set. If $A$  is a open set,
$X=A^c$  is a closed set. Hence, $\mathbb{R}$  and the empty set $\phi$  are both closed sets.
Then $\mathbb{R}$  and the empty set $\phi$  become open sets and closed sets, too.
In a general metric space we have to accept sets which have these two properties at a same time.
There are sets which are not open and closed. You may remember half open intervals.

The interior $X$  of a set $A$  is the maximum open set of the set $A$ . Of course, if $A$  is a open set, $X=A$ . In other words, $X$  is the open set including all open sets which belong to $A$ . We often write just like $X=\cup\left\{ Y\subset A | Y \mbox{ is open}  \right\}$ .

The closure $X$  of a set $A$  is the minimum closed set of the set $A$ . If $A$  is a closed set, $X=A$ . Correctly, $X$  is the minimum closed set which includes the set $A$ . We also often write $X=\cap\left\{ Y\supset A | Y \mbox{ is closed}  \right\}$ .

The boundary $X$ of a set $A$  is the set whose elements are the closure minus the interior.
Namely, the intersections of the neighborhoods of any elements in the boundary $X$  and $A$  is not empty,  and the intersections of the neighborhood and the complementary of $A$  is not empty, too.

These must be the most familiar definitions to you. However, you have to note that these are only based upon the definition of open sets.