Given a set $X\neq \emptyset$ it is well-known that the collection $\text{Top}(X)$ of all topologies on $X$ is a (complete) lattice with respect to $\subseteq$.

Let $0$ denote the smallest element of the lattice - in our case it is the indiscrete topology $\{\emptyset, X\}$. Given a lattice $L$ with a bottom element $0$ and $x\in L$ we say that $x^*\in L$ is a pseudocomplement if

- $x\wedge x^* = 0$, and
- if $x\wedge y = 0$ then $y\leq x^*$.

(In other words, $x^*$ is the largest element such that the infimum with $x$ is $0$.)

Question: Is there a set $X$ and $\tau\in\text{Top}(X)$ such that $\tau$ does not have a pseudocomplement?