My book in Analysis says:
Let $(S,d)$ be a metric space. $S$ is open in $S$ and the empty set $\emptyset$ is open in $S$.
Fair enough. But then they proceed...
A subset $E$ of $S$ is closed if its complement $S\setminus E$ is an open set. In other words, $E$ is closed if $E=S\setminus U$ where $U$ is an open set.
How is this not a contradiction? We have that $S=S\setminus\emptyset$. Therefore, $S$ should be closed. But $S$ is also open in $S$.
In fact, "open" and "closed" are not opposites. Indeed: in a space $S$, $S$ is both open and closed, as is $\emptyset$. A metric space is connected exactly when these are the only two subsets that are both closed and open.
Of course, many sets are neither closed nor open.
Somewhere (I think on this site) I read someone say that sets can be both open and closed because they aren't doors. It was a connection I'd never made before, but in everyday English, we do use "open" and "closed" as antonyms, like with doors being open or closed (but not both), bottles being open or closed (but not both), etc. This usage is totally irrelevant to the usage of "open" and "closed" in topology. A set can be both open and closed, or even neither open nor closed.