Here, my definition of Semiregular Space is $-$ a space which has a base of regularly open sets.
In Willard, it is written that -
A Semiregular $T_1$ space need not be $T_2$
However, Willard has not given an example, and I haven't been able to come up with an example myself. Any help would be appreciated!
Let $X=(0,1)^2\cup\{\langle 0,0\rangle,\langle 1,0\rangle\}$. Let $\mathscr{B}_0$ be the set of open balls in the Euclidean metric on $\Bbb R^2$ that are entirely contained in the open square $(0,1)^2$, and let
$$\begin{align*} \mathscr{B}&=\mathscr{B}_0\cup\left\{\{\langle 0,0\rangle\}\cup\left(\left(0,\frac23\right)\times\left(0,\frac1n\right)\right):n\in\Bbb Z^+\right\}\\ &\quad\quad\;\;\cup\left\{\{\langle 1,0\rangle\}\cup\left(\left(\frac13,1\right)\times\left(0,\frac1n\right)\right):n\in\Bbb Z^+\right\}\;; \end{align*}$$
$\mathscr{B}$ is a base for a $T_1$ topology on $X$ in which the subspace $(0,1)^2$ has its usual Euclidean topology. Clearly $X$ is not Hausdorff, since $\langle 0,0\rangle$ and $\langle 1,0\rangle$ do not have disjoint open nbhds. Finally, it’s not hard to check that the members of $\mathscr{B}$ are regular open sets.