Let me first make sure I have the correct definitions because my question will be about the difference about the two and there may be some massive confusion on my part.

A topological space $X$ is said to be **completely regular** or **Tychonoff** when it is Hausdorff and satisfies the following equivalent conditions:

For every $x \in X$ and closed $F \subseteq X$ such that $x\not\in F$, there exists a continuous $f\colon X\to\mathbb{R}$, which we can assume to have values in $[0,1]$, such that $f(x) = 0$ and $f|_F = 1$.

The map $X \to [0,1]^{C(X,[0,1])}$ taking $x\in X$ to the family $(f(x))_{f\in C(X,[0,1])}$ of its images under every continuous $f\colon X\to[0,1]$ defines a homeomorphism of $X$ to its image.

The Stone-Čech compactification map $X \to \beta X$ defines a homeomorphism of $X$ to its image.

There exists a compact [Hausdorff] space $K$ such that $X$ is homeomorphic to a subspace of $K$.

On the other hand, a (necessarily Hausdorff) topological space $X$ is said to be **functionally Hausdorff** (or **Urysohn**, but some people use this to mean something different, so it's probably best to avoid this terminology) when it satisfies the following equivalent conditions:

For every $x,y \in X$ such that $x\neq y$, there exists a continuous $f\colon X\to\mathbb{R}$, which we can assume to have values in $[0,1]$, such that $f(x) = 0$ and $f(y) = 1$.

The map $X \to [0,1]^{C(X,[0,1])}$ taking $x\in X$ to the family $(f(x))_{f\in C(X,[0,1])}$ of its images under every continuous $f\colon X\to[0,1]$ is injective.

The Stone-Čech compactification map $X \to \beta X$ is injective.

There exists a continuous injective map $X \to K$ with $K$ a compact [Hausdorff] space.

I note that example 91 (the “deleted Tychonoff corkscrew”) in Steen & Seebach's *Counterexamples in Topology* gives an example of a functionally Hausdorff space which is not completely regular, showing that the two notions are not equivalent.

Since until recently I thought these two notions were equivalent (I somehow thought that $X \to \beta X$ was automatically an embedding when it is injective), my goal is essentially to dispel the confusion I had; I first have to ask:

Question 0a:Is the above account correct? (Are the properties I claim to be equivalent indeed equivalent, and equivalent to standard definitions for the terms they claim to define?)

Every topological space $X$ has a **complete regularization** or **Tychonoff-ization**, namely a continuous map $X \to X'$ with $X'$ a completely regular space, such that every continuous map $X \to Y$ with $Y$ completely regular uniquely factors as $X \to X'\to Y$. (I.e., the functor $X \mapsto X'$ is left adjoint to the inclusion functor from the full subcategory of completely regular spaces to that of topological spaces.) This $X'$ can be defined as the image of the Stone-Čech compactification map $X \to \beta X$ with the subspace topology; in particular, $X \to X'$ is always surjective.

Question 0b:Is this still correct?

**Edit** (2019-06-08): Essentially the above description of complete regularization functor is given, under the name “**Tychonoff functor**” in the paper “The Tychonoff Functor and Related Topics” by T. Ishii, chapter 6 (p.203–243) in K. Morita & J. Nagata (eds.), *Topics in General Topology* (1989). So it would appear that it is correct.

Now I thought $X'$ was a quotient space of $X$. This can't be the case because, if what I wrote above is correct, the equivalence relation (“having the same image in $X'$”) is simply “having the same image under every continuous function $X\to\mathbb{R}$ (or equivalently $X\to[0,1]$)”, which is trivial for a functionally Hausdorff space, yet the latter is not necessarily completely regular.

But this is problematic because in section d-2 (“Higher Separation Axioms”, p.158–159) of the *Encyclopedia of General Topology* (Hart, Nagata & Vaughan eds.) one reads:

“To every space $X$ one can associate a Tychonoff space $Y$ as follows. Two points $x$ and $y$ in $X$ are equivalent if $f(x) = f(y)$ for all continuous real-valued functions $f$ on $X$. The corresponding

quotient space$Y$ is Tychonoff and the rings $C(X)$ and $C(Y)$ of real-valued continuous functions are isomorphic; the same holds for the rings $C^*(X)$ and $C^*(Y)$ of bounded real-valued continuous functions.”

It would seem to me that this assertion is contradicted by the existence of the aforementioned counterexample in Steen & Seebach.

Question 0c:Am I correct in believing that the above quote is in error? (Or did I miss some fine print or hidden assumption?)

Now assuming all of the above is correct, there are two natural questions which are left open:

Question 1a:Does every topological space $X$ have a “functional Hausdorffization” (or “Urysohnization”), namely, does the inclusion functor from the full subcategory of functionally Hausdorff spaces to that of topological spaces have a left adjoint? •Question 1b:If so, is it given by quotienting by the equivalence relation “$f(x) = f(y)$ for all continuous real-valued functions $f$ on $X$” or is there some subtlety?

**Edit** (2019-06-08): The paper “A universal factorization theorem in topology” (*Canad. Math. Bull.* **9** (1966) 201–207), by R. Sharpe, M. Beattie & J. Marsen defines an equivalence relation $T_{3/2}$ (it is not clear to me whether this is a typo for $T_{3\frac{1}{2}}$, and whether they may be under the same confusion as mentioned above) on any topological space by $x\,\mathbin{T_{3/2}}\,y$ when for every continuous $f\colon X\to [0,1]$ we have $f(x)=f(y)$, i.e., the relation defined above. Calling just $R$ this equivalence relation, according to this paper's main theorem, the crucial point in answering questions 1ab positively is to check that $R = \lim R$ in the notation of the paper, where $\lim R$ is the equivalence relation defined by $x\mathbin{(\lim R)}y$ when for every continuous $f\colon X\to Z$ with $Z$ such that $R$ is just equality (i.e., in our case, every functionally Hausdorff $Z$) we have $f(x)=f(y)$; but this is trivial: so it appears that questions 1ab have a positive answer.

Question 2a:Even if complete regularization is not given by a quotient, is there still a unique coarsest equivalence relation $R$ on any topological space such that $X/R$ is completely regular? •Question 2b:If so, can we describe $R$ concretely, and can we describe the natural continuous map $X' \to X/R$ (where $X'$ is complete regularization as defined above)?