Is This a Counterexample in Hausdorff Space?

Consider this topology on space X: $\tau=\{A\subset X:x_0\not\in A\text{ or }X-A\text{ is finite}\}$ where $x_0$ is a fixed point. Observe that this space is Hausdorff. However, singletons $\{u\}$ for u not $x_0$ are open. Aren’t all singletons supposed to be closed in Hausdorff?

Thanks.

Solution 1:

Sets are not doors. Sets can be open and closed at the same time. This in particular is true of $\{u\}$ with $u \neq x_0$ (open because $x_0 \notin \{u\}$; the complement of $\{u\}$ is open because $X\setminus\{u\}$ has a finite complement (namely $\{u\}$ again), so $X\setminus \{u\}$ is open, making the set $\{u\}$ itself closed). These are so-called clopen (closed-and-open) sets and they're not uncommon in topology. Closed just means the complement is open, remember? It's not the opposite of open, despite the naming.