Let the metric space $X = [0,1) \cup (2,3]$ with $d(x,y) = |x-y|$. Prove that $[0,1)$ is a closed subset. I know that it is a subset, because the set contains every limit point in the metric, but isn't $(2,3]$ closed as well? Wouldnt this be a violation of a closed set being closed if the complement is open?
The terminology is perhaps unfortunate. In topology, open and closed are not antonyms. A set can be
- open and not closed,
- closed and not open,
- open and closed, or
- neither open nor closed.
The empty set and the whole space are always both open and closed. You may have noticed that no other set in $\Bbb R$, $\Bbb R^2$, or $\Bbb R^3$ is both open and closed: this is because those spaces are "connected".
Edit: The last sentence takes things very badly out of historical context. Take with a grain of salt.