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

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.

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.

• Thank you for your great answer, Tyrone. Commented Sep 5 at 13:50