# "How many" separation axioms are there?

In my topology class, we've covered a number of separation axioms; specifically, we've covered the chain normal $$\implies$$ completely regular $$\implies$$ regular $$\implies$$ Hausdorff $$\implies$$ $$T_1$$. I also know that these axioms weren't discovered in order, resulting in situations like having $$T_{2\frac12}$$ or $$T_{3\frac12}$$. My question is, are the axioms I've listed (and the other common ones) the "only" separation axioms, or are there more that aren't as useful, and so aren't as commonly mentioned? Do we know how many axioms could exist, or do we know if we've found all of them?

• Counterexamples in Topology is a great book that spends time on these. Apr 14 at 19:23
• en.wikipedia.org/wiki/Separation_axiom?wprov=sfti1 The Wikipedia page on separation axioms is a good start. Apr 14 at 19:41

"Separation axiom" is an informal notion, not something that's precisely defined. One could write down arbitrarily many logical conditions that distinguish topologies into finer and finer classes (until each class is not provably larger than a single homeomorphism class), but we probably wouldn't consider all these conditions to be "about separation" or to be useful in proving other properties of interest.

I would say there is ton of separation axioms, but most of them are not important. Also, they do not form a mere collection or a chain of properties – they can be organized in various meaningful ways. Let me illustrate this on some examples.

One can build a preliminary version of a “periodic table” of separation axioms by considering two aspects: what we are separating (points, points and closed sets, closed sets) and how we are separating (by disjoint neighborhoods, by neighborhoods with disjoint closure, by function, …). I personally call the groups of axioms corresponding to what we are separationg by “tier 2, tier 3, tier 4” axioms. Tier 2 includes $$T_2$$/Hausdorff, $$T_{2\frac{1}{2}}$$/Urysohn, functionally Hausdorff. The corresponding axioms in tier 3 are $$T_3$$/regular and $$T_{3\frac{1}{2}}$$/completely regular. Note that the separation by neighborhoods and the separation by closed neighborhoods collapsed here. Tier 4 includes $$T_4$$/normal – here the three ways of separation collapsed.

Let's come back to $$T_0$$ and $$T_1$$. I put them to tier 1. Note that applying $$T_0$$ or $$T_1$$ separation to closed sets and $$T_0$$ separation to a point and a closed set is trivial. But the $$T_1$$ separation applied to a point and a closed set reads as follows: for every $$x ∉ F$$ there is $$U$$ such that $$x ∉ U ⊇ F$$, or equivalently $$x ∈ \overline{\{y\}}$$ iff $$y ∈ \overline{\{x\}}$$. This axiom is called symmetry, and one may observe that $$T_1$$ is exactly the conjunction of $$T_0$$ and symmetry. Let's put it to tier 1 as well.

Note that the “bare” axioms from tier 3 do not imply the corresponding axioms in tier 2 since tier 2 implies $$T_1$$. But since tier 3 implies symmetry, it is enough to add $$T_0$$ to have the implication. So every axiom in tier 3 has the bare and the $$T_0$$ variant. This is the distinction between regular and $$T_3$$ (but both competing conventions of how to assign the names to the properties are used). Similarly for the implication from tier 4 to tier 3, it is enough to add symmetry, and for tier 4 $$\implies$$ tier 2 we need to add $$T_1$$. (Note that this means that there are four variants of normality: bare normality, symmetric normality, $$T_0$$ normality, and $$T_1$$ normality. But practically this is not of much importance.)

There are more ways how to enrich our table of separation axioms. Essentially, all separation axioms are stable under sums and none are stable under quotients. Tiers 1, 2, 3 are also stable under subspaces and products, but this is not the case for tier 4. So we may consider productively normal and heredirily normal (also called completely normal or $$T_5$$). Or we may consider more means of separation, e.g. the perfect separation: sets $$A, B ⊆ X$$ are perfectly separated if there is a continuous map $$f: X \to [0, 1]$$ such that $$A = f^{-1}(0)$$ and $$B = f^{-1}(1)$$. The tier 4 variant is called perfect normality or $$T_6$$ (as it turns out, the property is hereditary, and so implies $$T_5$$). One may also consider perfectly Hausdorff and pefectly regular (although in the latter case, one has to be careful: if we demand perfectness on the closed set, we already obtain $$T_6$$, and if we demand it on the point, we obtain “just” the conjunction of complete normality and perfect Hausdorffness). Another mean is separation by a clopen set (stronger than functional separation, but incomparable with perfect separation) – we obtain totally separated, zero-dimensional, and strongly zero-dimensional/ultranormal. I think I've even once seen called them $$D_2$$, $$D_3$$, $$D_4$$. $$D_1$$ would be hereditarily disconnected, even though it doesn't exactly fit the scheme (but it implies $$T_1$$). Also note that totally disconnected is inconsitently used both for $$D_1$$ and $$D_2$$.

To answer your question, note that picking a space $$S$$ and its subsets $$S_0, S_1$$, we may define the corresponding mean of separation: $$A, B ⊆ X$$ are $$(S, S_0, S_1)$$-separated is there is a continuous map $$f:X \to S$$ such that $$f[A] ⊆ S_0$$ and $$f[B] ⊆ S_1$$. So in theory, you can pick your favourite space and sets and make your own separation axiom. Note that the standard axioms (but using perfect separation) are of this form.

There are much more things I didn't mention: collectionwise Hausdorffness and normality related to paracompactness, axioms relative to other axioms (preregularity is $$T_2/T_0$$, you can substract $$T_0$$ from any property), relative axioms for subspaces (two points in $$X ⊆ Y$$ separated by disjoint neighborhoods in $$Y$$), very weak axioms like $$T_{\frac{1}{2}}$$ and $$T_D$$, point-free axioms like subfitness, etc.