Regular $G_{\delta}$ spaces and complete regularity

Question

Let $X$ be a topological space.

$X$ is regular if given a proper closed set $C$ in $X$ and a point $x \in X \setminus C$, then exist $A$ and $B$ disjoint open sets in $X$ such that $x \in A$ and $C \subseteq B$.

$X$ is normal if given $C$ and $D$ disjoint closed sets in $X$, then exist $A$ and $B$ disjoint open sets in $X$ such that $C \subseteq A$ and $D \subseteq B$.

Equivalently, this is known as Urysohn's Lemma, $X$ is normal if given $C$ and $D$ disjoint closed sets in $X$, then exists $f : X \rightarrow [0, 1]$ continuous function such that $C \subseteq f^{-1}(\{0\})$ and $D \subseteq f^{-1}(\{1\})$.

$X$ is completely regular if given a proper closed set $C$ in $X$ and a point $x \in X \setminus C$, then exists $f : X \rightarrow [0, 1]$ continuous function such that $C \subseteq f^{-1}(\{0\})$ and $f(x) = 1$.

$X$ is a $G_{\delta}$ space if each closed set in $X$ is a countable intersection of open sets in $X$.

I know how normality is used in the proof of Urysohn's Lemma. And, this doesn't hold for regular spaces. I also know examples of regular spaces that aren't completely regular, not even functionally Hausdorff.

The thing is that I wanted to find examples of $T_3$ (regular and $T_0$) $G_{\delta}$ spaces that aren't completely regular. However, I haven't been able to find any. Does anyone know any examples of such, or regular + $G_{\delta}$ space implies completely regular?

Note: in $\pi$-Base there are no examples of regular + $G_{\delta}$ space + $\neg$completely regular spaces, and there are no examples of regular + Countable pseudocharacter (each point is a $G_{\delta}$ set) + $\neg$completely regular spaces.

Answer 1

A development for a space $X$ is a sequence $\{\mathcal{U}_n\mid n\in\mathbb{N}\}$ of open coverings such that for each $x\in X$ the family $\{st(x;\mathcal{U}_n)\mid n\in\mathbb{N}\}$ is a local base at $x$, where $st(x;\mathcal{U}_n)=\bigcup\{U\in\mathcal{U}_n\mid x\in U\}$. A regular Hausdorff space admitting a development is called a Moore space.

It is clear that each Moore space is perfect (i.e. a $G_\delta$-space), and it can be shown that each Moore space is subparacompact and has a $G_\delta$-diagonal.

In the paper

F.B. Jones, Moore spaces and uniform spaces, Proc. Amer. Math. Soc. 9 (1958), 483-486.

F.B. Jones gives an example of a connected, locally connected, Moore space which is not completely regular.

Later, S. Armentrout modified Jones's example so as to construct an infinite Moore space on which each continuous, real-valued function is constant. Such a space is clearly not completely regular. The construction is short and found in

S. Armentrout, A Moore Space on Which Every Real-Valued Continuous Function Is Constant, Proc. Amer. Math. Soc. 12 (1961), 106-109.

Further examples of such spaces were later produced by several mathematicians. For example in the paper

V. Tzannes, Two Moore Spaces on Which Every Continuous Real-Valued Function is Constant, Tsukuba J. Math. 16 (1) (1992), 203-210.

contains several such spaces, as well as a short bibliography pointing to other constructions.