Separation Axioms Among US

Steven Clontz
University of South Alabama

2025 September 06
Set Theory and Topology in Messina

But first, a word from our sponsor!

• The Team-Based Inquiry Learning Resource Library (TBIL.org) is a repository of free and open-source evidence-based materials for active learning Precalculus, Calculus, and Linear Algebra classrooms.
• That work (and this visit to STTM communicating it) was made possible through the support of National Science Foundation Award #2011807.

Seeking: Special Session organizers for SumTopo 2026

SumTopo 2026 will be in Split, Croatia from July 13-17! We're seeking organizers for a special session in General and Set-Theoretic Topology. Email steven@clontz.org to volunteer or visit Clontz.org/SumTopo to learn more.

An apology:

This morning's schedule.

• 9:00am, Jocelyn: "Games!"
• 9:25am, Steven: who cares
  (you are here)
• 9:50am, Michał: "Games!"

I usually prefer topological games too, but there's a reason I'm switching gears today!

PSA: Careful that you can unlock your bathroom doors...

Don't be a victim like Will Brian!

Now back to your scheduled programming...

Abstract

A standard introductory result is that Hausdorff spaces have the property US, that is, each convergent sequence has a unique limit. Here we will explore several existing and new characterizations of separation axioms that are strictly weaker than Hausdorff but strictly stronger than US.

Based upon joint work [0] with Marshall Williams.

[0] C., Williams, M. Separation Axioms Among US. Topology and its Applications, Volume 375, 2025.

Where our story begins.

In [1], Wilansky published a systematic overview of two properties strictly between and : KC (Kompacts are Closed) and US (converging Sequences have Unique limits).

One may show that

with no arrows reversing.

[1] Wilansky A. Between and . Amer Math Monthly. 1967;74:261-6. Available from: https://doi.org/10.2307/2316017.

But why am I interested in this?

My dirty secret: I'm only a part-time topologist these days.

A major part of my active scholarly program is focused on the digital and social infrastructure of mathematics research [2], and I have a particular interest in databases of research mathematics.

And my favorite database? The -Base community database of topological counterexamples, of course.

[2] C. Database-Driven Mathematical Inquiry and the -Base Model for Small Semantic Databases. Proceedings of LMFDB, Computation, and Number Theory (LuCaNT). Contemporary Mathematics. (To appear).

What is needed to contribute properties like KC and US to the -Base?

• Notability
  • KC (P100) and US (P99) are indeed properties recognized in the literature.
• Connections
  • The theorems , , and connect these properties to others known to -Base.
• Counter-examples
  • Spaces (e.g.) Cocountable topology on (S17), Square of one-point compactificiation of (S31), and Cofinite topology on (S15) witness that each implication does not reverse.

Done! Now what?

Part of what's fun about contributing to -Base today is that there's still so much ground to cover to catch up to the current state of the literature.

• It seems Patrick Rabau was the first to observe in 2021 [3] that the wH (Weak Hausdorff, P143) property lies strictly between KC and US.
• Madison and Lawson define kH (k-Hausdorff) in [4], and showed it lives between and KC.
• Rezk also defines kH in [5], and showed it is implied by wH.

[3] Rabau, P. Relationship between weak Hausdorff and US properties. Math StackExchange (2021).

[4] Madison, B., Lawson, J. Quotients of k-semigroups. Semigroup Forum 9 (1974): 1-18.

[5] Rezk, C. Compactly generated spaces. nLab (2018).

Cleaning up k-Hausdorff

On Math StackExchange [6], Rabau and I took a careful look at the kH property. In both [4] and [5], a space was defined to be k-Hausdorff provided that its diagonal is k-closed.

The trick is that there are two inequivalent definitions of k-closed in the literature. Let's say a set is -closed provided its intersection with every compact subspace is closed, and -closed provided its intersection with the image of any compact Hausdroff space is closed. We then obtain the respective definitions for what we will now call H (-Hausdorff, P170) and H (-Hausdorff, P171)

[6] C, and Rabau, P. How are k-Hausdorff and weakly Hausdorff distinct? Math StackExchange (2023).

Putting it together

As it turns out, we have

with no arrows reversing. In fact:

[7] C. Non-Hausdorff Properties. arXiv (2024). (Under revision to resubmit, I promise!)

Generalizing US

Let's dig into another definition for :

For every compact Hausdorff space , continuous map , and points with , there exist open neighborhoods of with .

Let and we easily see why .

What about other types of sequences?

In the Carolinas virtual topology seminar (soon to be revived by Lynne Yengulalp and Jocelyn Bell) I presented this work. Alan Dow asked if I'd considered longer sequences.

I considered this question rather SUS (Strongly Uniquely Sequential), but wrote it up on Math StackExchange anyway [8]. The post was popular, likely because of all the inside jokes ඞ I baked into it. But the attention eventually led user @MW (my now-coauthor Marshall Williams) to find the answer.

[8] C., Williams, M. "Is there anyone among us who can identify a certain SUS space?" Math StackExchange (2023).

Transfinite sequences

Define a transfinite sequence to be a function from a limit ordinal into a topological space. Note that every -length (transfinite) sequence is continuous, but longer transfinite sequences need not be.

A limit of a transfinite sequence is a point such that every neighborhood contains a final tail of the sequence; the transfinite sequence is said to converge to this limit.

UR and UCR

This suggests two "SUS" candidates:

• A space is UR (Unique Radial limits) provided the limit of every converging transfinite sequence is unique.

• A space is UCR (Unique Continuously-Radial limits) provided the limit of every converging continuous transfinite sequence is unique.

We see immediately that

(Our paper also considers a property UOK intermediate to UR & UCR, which we omit here.)

Counterexamples

• The space with its endpoint doubled (S37) is US but not UCR.

• Take a compact, Hausdorff, sequentially discrete (convergent sequences are eventually constant, P167) with a non-isolated point (e.g. , S108). Doubling this point produces a UCR but not space.

  • To see this, note sequentially discrete implies all converging continuous transfinite sequences are eventually constant, and thus (given ) have a unique limit.

UCR but not UR

In fact, this example fails to be UR. Note that every non-isolated point in a compact Hausdorff space is radially accessible, that is, there is a transfinite sequence (not necessarily continuous) of distinct points converging to it.

• Idea: take a point, forbid a neighborhood around it whose closure misses the isolated point, take a second (non-forbidden) point, rinse, repeat...

So, take the doubled point, and take a transfinite sequence of distinct points converging to it in : this witnesses UR.

Where does UCR fit?

Just as witnesses the proof that , we may consider to convince ourselves that . Therefore:

Where does UR fit?

Interestingly, it doesn't! At least no more than we've already shown. Indeed, we can show

with no arrows reversing or missing.

Proof

We'll ignore the (heretofore undefined) lH, sH, RC, and UOK properties. So I owe you examples which are but not UR, and UR but not even .

but not UR

The "standard" example of a (which may be characterized by compact subspaces are all Hausdorff) which fails is not UR: consider the co-countable topology on an uncountable set (S17). Here compacta are finite and thus . But non-trivial transfinite sequences of uncountable length converge to every point of the space, violating UR.

UR but not

To obtain an example which is UCR which failed UR, we "cheated" by doubling a point of the remainder in . Note that this space also fails : consider the natural maps from to this space that differ only on this doubled non-isolated point.

So to obtain a UR space which fails , we want to use the same trick to violate , but avoid transfinite sequences with non-unique limits.

This can be done by doubling every point of . In this space, converging transfinite sequences are either eventually constant, or eventually live in one of the two copies of , and thus converge within that copy to a unique limit.

My pitch

While sub-Hausdorff separation axioms are worthy of study independently, I'm more interested in pitching the methodology that inspired this investigation.

Mary Ellen Rudin once said in her review of Counterexamples in Topology (paraphrased):

Topology is a dense forest of counterexamples. A usable map of the forest is a fine thing.

We hope that -Base can be this map for the modern era. Not only does it help mathematicians find answers, it is also an incredibly useful tool for discovering questions, especially for student/early-career folks. Contributing to the database also builds "real world" (cough) tech skills for students who may end up outside of research.

Questions?

Thanks for listening! Find me at Clontz.org, and check out TBIL.org to bring active learning into your early-college classrooms!