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 $text{T}_3 nRightarrow text{T}_{3frac{1}{2}}$. 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 $text{Completely Regular}$?

DanielWainfleet · Answer

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 $pne q.$
Let $U^*= {uin U: {p,q}subseteq ulor {p,q}cap u=emptyset}.$
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.

Taumatawhakatangihangakoauauot · Answer

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.

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

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