I recently saw a proof that in a topological group, the closure of a point equals the intersection of all its neighbourhoods.
Intuitively, it seemed to me that this might hold in general (i.e. in other topological spaces that are not topological groups). So what are some counterexamples? Moreover, if the equality doesn't hold in general, does one of the inclusions always hold? Is there a larger category of spaces in which equality holds?
(I self-answer because even though I found an answer myself after some thought, I think it's an interesting question, and other users might contribute other enlightening examples.)
Consider the following topological space, where the letters are points and the ellipses are the open sets (since these sets are linearly ordered by inclusion, this is indeed a topology):
Then the closure of $\{B\}$ is $\{B, C\}$ but the intersection of all neighbourhoods of $B$ is $\{A, B\}$. So neither is included in the other.
In general, I believe that in any non-symmetric (non-$R_0$) space there will be a point for which these two things are not equal.