Example of $\text{Regular}$ non-$T_0$ space which is not $\text{Completely Regular}$?

Question

Note: I'm assuming that the definition of Regular and Completely Regular spaces do not require them to be $T_0$.

While studying Topology, I've found examples that show that $\text{T}_3 \nRightarrow \text{T}_{3\frac{1}{2}}$. However, I've not been able to find an example of a space which shows that $\text{Regular} \nRightarrow \text{Completely Regular}$ without assuming $T_0$ (and thus going back to the case above). Even Steen and Seebach has no examples of such spaces.

So, does there exist a $\text{Regular}$ non-$T_0$ space which is not $\text{Completely Regular}$?

Answer 1

Take your favorite example of a regular not completely regular T₀-space $X$, take two points $\spadesuit , \clubsuit$ not belonging to $X$, and give $Y = X \cup \{ \spadesuit , \clubsuit \}$ the topology generated by the base consisting of the topology on $X$ and $\{ \spadesuit , \clubsuit \}$.

• This is not T₀ because $\spadesuit , \clubsuit$ have exactly the same neighborhoods.
• This space is still regular because it is the disjoint union (topological sum) of two regular spaces, $X$ and the indiscrete space $\{ \spadesuit , \clubsuit \}$.
• This space is not completely regular because the subspace $X$ is not completely regular.

Answer 2

Let $U$ be a regular, not completely regular topology on $X.$ Since $U$ is not completely regular, $X$ has more than 1 member. Choose $p,q \in X$ with $p\ne q.$

Let $U^*= \{u\in U: \{p,q\}\subseteq u\lor \{p,q\}\cap u=\emptyset\}.$

Show that $U^*$ is a regular topology on $X.$

Now $U^*\subset U$ so if $U^*$ were completely regular then $U$ would be completely regular.