Call a topological space $\langle X,\mathscr{O}\rangle$ regular iff it is both $T_0$ and $T_3$: for every point $x\notin A$, where $A$ is a closed subsets of $X$, there are open and disjoint sets $V$ and $U$ such that $x\in V$ and $A\subseteq U$.

A $\langle X,\mathscr{O}\rangle$ space is well-based iff in every its point there is a local basis which is linearly ordered by $\subseteq$ relation.

Could you give me an example of a regular space which is not well-based?

Call a topological space $\langle X,\mathscr{O}\rangle$ regular iff it is both $T_0$ and $T_3$: for every point $x\notin A$, where $A$ is a closed subsets of $X$, there are open and disjoint sets $V$ and $U$ such that $x\in V$ and $A\subseteq U$.

A $\langle X,\mathscr{O}\rangle$ space is linear-based iff in every its point there is a local basis which is linearly ordered by $\subseteq$ relation.

Could you give me an example of a regular space which is not linear-based?