Disclaimer: By regular (completely regular) we mean regular (completely regular) and $T_{0}$

In general topology, there are two different kinds of topological spaces. There are the topological spaces that satisfy higher separation axioms such as the 3 dimensional space that we live in; when most people think of general topology (especially analysts and algebraic topologists), they usually think of spaces which satisfy higher separation axioms. On the other hand, there are topological spaces which only satisfy lower separation axioms; these spaces at first glance appear very strange since sequences can converge to multiple points. They feel much different from spaces which satisfy higher separation axioms. These spaces include the Zariski topology, finite non-discrete topologies, and the cofinite topology. Even spaces that set theorists consider such as the ordinal topology on a cardinal $\kappa$ or the Stone-Cech compactication $\beta\omega$ satisfy higher separation axioms; after all, $\beta\omega$ is the maximal ideal space of $\ell^{\infty}$. The general topology of lower separation axioms is a different field of mathematics than the general topology of higher separation axioms.

However, can we in good conscience formally draw the line between the lower separation axioms and the higher separation axioms or is the notion of a higher separation axiom simply an informal notion? If there is a line, then where do we draw the line between these two kinds of topological spaces?

As the sole owner of a silver badge in general topology on mathoverflow, I declare that the axiom *complete regularity* is the place where we need to draw the line between the lower separation axioms and the higher separation axioms. I can also argue that complete regularity is correct cutoff point by appealing to an authority greater than myself; the American Mathematical Society’s MSC-classification (the authority on classifying mathematics subjects) also delineates the lower separation axioms and the higher separation axioms at around complete regularity:

`54D10-Lower separation axioms ($T_0$–$T_3$, etc.)`

`54D15-Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)`

Let me now give a few reasons why complete regularity is the pivotal separation axiom.

Hausdorffness is not enough. We need at least regularity.

Hausdorff spaces are appealing to mathematicians because Hausdorff spaces are precisely the spaces where each net (or filter) converges to at most one point. However, the condition that every net converges to at most one point should not be enough for a space to feel like it satisfies higher separation axioms. Not only do I usually want filters to converge to at most one point, but I also want the closures of the elements in a convergent filter to also converge. However, this condition is equivalent to regularity.

$\mathbf{Proposition}:$ Let $X$ be Hausdorff space. Then $X$ is regular if and only if whenever $\mathcal{F}$ is a filter that converges to a point $x$, the filterbase $\{\overline{R}\mid R\in\mathcal{F}\}$ also converges to the point $x$.

The next proposition formulates regularity in terms of the convergence of nets. The intuition behind the condition in the following proposition is that for spaces that satisfy higher separation axioms, if $(x_{d})_{d\in D},(y_{d})_{d\in D}$ are nets such that $x_{d}$ and $y_{d}$ get closer and closer together as $d\rightarrow\infty$, and if $(y_{d})_{d\in D}$ converges to a point $x$, then $(x_{d})_{d\in D}$ should also converge to the same point $x$.

$\mathbf{Proof:}$ $\rightarrow$ Suppose that $(x_{d})_{d\in D}$ does not converge to $x$. Then there is an open neighborhood $U$ of $x$ where $\{d\in D\mid x_{d}\not\in U\}$ is cofinal in $D$. Therefore, there is some open set $V$ with $x\in V\subseteq\overline{V}\subseteq U$. Therefore, let $U_{d}=(\overline{V})^{c}$ whenever $d\in D$ and $U_{d}$ be an arbitrary set otherwise. Then whenever $y_{d}\in U_{d}$ for each $d\in D$, the set $\{d\in D\mid y_{d}\not\in U\}$ is cofinal in $D$. Therefore, $(y_{d})_{d\in D}$ does not converge to $x$ either.

$\leftarrow$ Suppose now that $X$ is not regular. Then there is an $x\in X$ and an open neighborhood $U$ of $x$ such that if $V$ is an open set with $x\in V$, then $V\not\subseteq U$. Therefore, let $D$ be a directed set and let $U_{d}$ be an open neighborhood of $x$ for each $d\in D$ such that for all open neighborhoods $V$ of $x$ there is a $d\in D$ so that if $e\geq d$, then $U_{d}\subseteq V$. Then let $x_{d}\in\overline{U_{d}}\setminus U$ for all $d\in D$. Then $(x_{d})_{d\in D}$ does not converge to $x$. Now suppose that $V_{d}$ is a neighborhood of $x_{d}$ for each $d\in D$. Then for each $d\in D$, we have $V_{d}\cap U_{d}\neq\emptyset$. Therefore, let $y_{d}\in V_{d}\cap U_{d}$. Then $(y_{d})_{d\in D}$ does converge to $x$. $\mathbf{QED}$.

Complete regularity is closed under most reasonable constructions

If there is a main separation axiom that draws the line between higher separation axioms and lower separation axioms, then this main separation axiom should be closed under constructions such as taking subspaces and taking arbitrary products. Since every completely regular space is isomorphic to a subspace $[0,1]^{I}$, the crossing point between lower and higher separation axioms should be no higher than complete regularity.

Not only are the completely regular spaces closed under taking products and subspaces, but the completely regular spaces are also closed under taking ultraproducts, the $P$-space coreflection, box products and other types of products, and various other constructions. Since we want our main separation axiom to be closed under most reasonable standard constructions and no lower than regularity, regularity and complete regularity are the only two candidates for our main separation axiom. We shall now find out why complete regularity is a better candidate than regularity for such a separation axiom.

Completely regular spaces can be endowed with richer structure

The completely regular spaces are precisely the spaces which can be given extra structure that one should expect to have in a topological space.

While a topological space gives one the notion of whether a point is touching a set, a proximity gives on the notion of whether two sets are touching each other. Every proximity space has an underlying topological space. Proximity spaces are defined in terms of points and sets with no mention of the real numbers, but proximity spaces are always completely regular. Furthermore, the compatible proximities on a completely regular space are in a one-to-one correspondence with the Hausdorff compactifications of the space.

$\mathbf{Theorem:}$ A topological space is completely regular if and only if it can be endowed with a compatible proximity.

The notion of a uniform space is a generalization of the notion of a metric space so that one can talk about concepts such as completeness, Cauchy nets, and uniform continuity in a more abstract setting. A uniform space gives one the notion of uniform continuity in the same way the a topological space gives one the notion of continuity. The definition of a uniform space is also very set theoretic, but it turns out that that every uniform space is induced by a set of pseudometrics and hence completely regular.

$\mathbf{Theorem:}$ A topological space is completely regular if and only if it can be endowed with a compatible uniformity.

For example, it is easy to show that every $T_{0}$-topological group can be given a compatible uniformity. Therefore, since the topological groups can always be given compatible uniformities, every topological group (and hence every topological vector space) is automatically completely regular.

Complete regularity is the proper line of demarcation between low and high separation axioms since the notions of a proximity and uniformity (which capture intuitive notions related to topological spaces without referring to the real numbers) induce precisely the completely regular spaces.

The Hausdorff separation axiom generalizes poorly to point-free topology

I realize that most of my readers probably have not yet been convinced of the deeper meaning behind point-free topology, but point-free topology gives additional reasons to prefer regularity or complete regularity over Hausdorffness.

Most concepts from general topology generalize to point-free topology seamlessly including separation axioms (regularity, complete regularity, normality), connectedness axioms (connectedness, zero-dimensionality, components), covering properties (paracompactness,compactness, local compactness, the Stone-Cech compactification), and many other properties. The fact that pretty much all concepts from general topology extend without a problem to point-free topology indicates that point-free topology is an interesting and deep subject. However, the notion of a Hausdorff space does not generalize very well from point-set topology to point-free topology. There have been a couple attempts to generalize the notion of a Hausdorff space to point-free topology. For example, John Isbell has defined an I-Hausdorff frame to be a frame $L$ such that the diagonal mapping $D:L\rightarrow L\oplus L$ is a closed localic mapping ($\oplus$ denotes the tensor product of frames). I-Hausdorff is a generalization of Hausdorffness since it generalizes the condition “$\{(x,x)\mid x\in X\}$ is closed” which is equivalent to Hausdorffness. Dowker and Strauss have also proposed several generalizations of Hausdorffness. You can read more about these point-free separation axioms at Karel Ha’s Bachelor’s thesis here. These many generalizations of the Hausdorff separation axioms are not equivalent. To make matters worse, I am not satisfied with any of these generalizations of Hausdorffness to point-free topology.

It is often the case that when an idea from general topology does not extend very well to point-free topology, then that idea relies fundamentally on points. For example, the axiom $T_{0}$ is completely irrelevant to point-free topology since the axiom $T_{0}$ is a pointed concept. Similarly, the axiom $T_{1}$ is not considered for point-free topology since the notion of a $T_{1}$-space is also fundamentally a pointed notion rather than a point-free notion. For a similar reason, Hausdorffness does not extend very well to point-free topology since the definition of Hausdorffness seems to fundamentally rely on points.

Just like in point-set topology, in point-free topology there is a major difference between the spaces which do not satisfy higher separation axioms and the spaces which do satisfy higher separation axioms. The boundary between lower separation axioms and higher separation axioms in point-set topology should therefore also extend to a boundary between lower separation axioms and higher separation axioms in point-free topology. Almost all the arguments for why complete regularity is the correct boundary between lower and higher separation axioms that I gave here also hold for point-free topology. Since Hausdorffness is not very well-defined in a point-free context, one should not regard Hausdorffness as the line of demarcation between lower separation axioms and higher separation axioms in either point-free topology or point-set topology.

Conclusion

Spaces that only satisfy lower separation axioms are good too.

While completely regular spaces feel much different from spaces which are not completely regular, spaces which satisfy only lower separation axioms are very nice in their own ways. For example, non $T_{1}$-spaces have a close connection with ordered sets since every non-$T_{1}$-space has a partial ordering known as the specialization ordering. I do not know much about algebraic geometry, but algebraic geometers will probably agree that spaces which only satisfy the lower separation axioms are important. Frames (point-free topological spaces) which only satisfy lower separation axioms are also very nice from a lattice theoretic point of view; after all, frames are precisely the complete Heyting algebras.

The underappreciation for complete regularity

The reason why Hausdorffness is often seen as a more important separation axiom than complete regularity is that Hausdorffness is easy to define than complete regularity. The definition of Hausdorffness only refers to points and sets while complete regularity refers to points, sets, and continuous real-valued functions. Unfortunately, since the definition of complete regularity is slightly more complicated than the other separation axioms, complete regularity is not often given the credit it deserves. For example, in the hierarchy of separation axioms, complete regularity is denoted as $T_{3.5}$. It is not even given an integer. However, Hausdorffness is denoted as $T_{2}$, regularity is denoted as $T_{3}$ and normality is denoted as $T_{4}$. Furthermore, when people often mention separation axioms they often fail to give complete regularity adequate attention. When discussing separation axioms in detail, one should always bring up and emphasize complete regularity.

In practice, the Hausdorff spaces that people naturally comes across are always completely regular. I challenge anyone to give me a Hausdorff space which occurs in nature or has interest outside of general topology which is not also completely regular. The only Hausdorff spaces which are not completely regular that I know of are counterexamples in general topology and nothing more. Since all Hausdorff spaces found in nature are completely regular, complete regularity should be given more consideration than it is currently given.