Topology, Set Theory, and the -Base

59th Spring Topology and Dynamics Conference

Steven Clontz, 2026 March 13

https://clontz.org | ScholarLattice Event Page

Thanks to the session organizers!

No actually, thank YOU!

It's great to have such great research colleagues.

Topology, Set Theory, and the -Base

Abstract. The -Base was recognized in Fall 2025 as the highest-voted crowdsourced math project on Terence Tao's MathOverflow list. While much can be done by simply modeling Objects/Spaces, Properties, and Theorems, without a notion of set theory and cardinality, we quickly find limitations, for example:

• Several "open questions" on -Base are equivalent to the Continuum Hypothesis
• Thirteen properties on -Base are just different cardinalities, with explicit theorems written to connect them.

We will discuss the current plan to incorporate results from set theory into the -Base, and seek input from potential users.

What is a -Base?

• A collection of nonempty open sets such that for every nonempty open set there exists a such that .
• However, we've ruined your ability to find that out via Google, because it's also a reasonably popular website for querying knowledge about topological spaces, properties, and theorems.
  • Check it out at Topology.pi-Base.org

Brief aside:

Developing the -Base as "scholarship"

• Even though I've put in a hopefully-successful application for full professor and can now do whatever I want (???), I think it's important to consider how academic labor "counts".
  • So how can folks make the -Base "count"?
• I have two traditional papers inspired by -Base work, a survey on -not- properties [C., to appear], and a study on the generalization of "US" (Unique Sequential limits) to transfinite sequences [C. and Williams, 2025].

Developing the -Base as "scholarship", cont.

• LuCaNT is a conference series on mathematical databases, computation, and number theory.
  • "But mostly number theory and the LMFDB"
• But I did contribute to LuCaNT's special issue of Contemporary Mathematics about the "-Base model" itself: Objects (from any category), Properties, and Theorems.
• And in that paper, I pointed out another problem with making the -Base "count", in a different sense...

Set Theory in the -Base

So how can we expand the -Base model to "know" about set theory (in so-much as it "knows" about topology at least)?

• Restrictions/considerations:
  • Implementation details: pretty sure there isn't an "ordinals/cardinals" Javascript library out there
  • Contributions: how will people contribute results involving set theory to the database?
  • Website: how will this data be presented to end-users?
  • Funding: the model shouldn't be applicable to / useful for just topology

Modeling Cardinals

• Expressing cardinals:
  • Finite: 0, 1, 2, ... for
  • Infinite: aleph 0, aleph 1, ... for
  • Also: beth 1, beth 2, ... for (where )
• What about ordinals?
  • Problem: need to know things like aleph n < beth m () for .
  • Could support e.g. aleph omega0+2 for . But need to write (find?) a library to know 7 < omega0+2. (Is this necessary for MVP?)
• Cardinal characteristics of the continuum.
  • Maybe: char_b, char_t, char_d, ... for
  • Need to manually model e.g. .

Properties with cardinal values

We'll need to distinguish (existing) properties with boolean values:

uid: P000089
name: Fixed point property
# values: boolean  # new key to keep existing functionality

... and hydrate others:

uid: P000026
name: Separable
# values: boolean

uid: P000209
name: Density $\leq\mathfrak c$
# values: boolean

as new properties with cardinal values:

uid: P000xyz
name: Density
values: cardinal

Cardinal-valued traits (space/property pairs)

Now we can replace such assertions:

space: S000222  # Product topology on $\omega^{2^\mathfrak{c}}$
property: P000026  # Separable
value: false  # actually this is deduced automatically today

space: S000222  # Product topology on $\omega^{2^\mathfrak{c}}$
property: P000209  # Density $\leq\mathfrak c$
value: true

with more expressive ones:

space: S000222  # Product topology on $\omega^{2^\mathfrak{c}}$
property: P000xyz  # Density
value: beth 1

Now, theorems!

Consider the following theorem.

uid: T000440
if:
  P000180: true  # Hereditarily separable
then:
  P000026: true  # Separable

The trivial generalization is "if every subspace has a dense subset of cardinality , then the space itself has a dense subset of cardinality ".

Supporting a seems most reasonable to me today:

uid: T000xxx
if:
  P000yyy: kappa  # Hereditary density
then:
  P000xyz: kappa  # Density

Supporting notation like kappa+ () seems reasonable, but we lack the machinery for understanding e.g. .

We also have this:

uid: T000404
if:
  and:
    - P000079: true  # Sequential
    - P000099: true  # US
    - P000209: true  # Density $\leq\mathfrak c$
then:
  P000163: true  # Cardinality $\leq\mathfrak c$

Here I think we need to support leq:

uid: T000404
if:
  and:
    - P000079: true  # Sequential
    - P000099: true  # US
    - P000xyz: leq beth 1  # Density
then:
  P000zyx: leq beth 1  # Cardinality

Of course, is a sequential US space which has density but cardinality .

Implementation as software

So this means we just need a to program a few small things:

• A Cardinality class that understands , , , and "custom cardinals" e.g. .
  • Also, Cardinality.leq:boolean so new Cardinality("aleph 3") <= new Cardinality("beth 4") returns true and new Cardinality("char_b") < new Cardinality("aleph 1") returns... false? "undecidable"?
• A string parser that reads leq beth 3 and abstracts it to a Cardinality that represents .
• A deduction engine that understands the kappa placeholder in our theorems.
• That... that's all?!

Ultimately...

With this infrastructure, we could not only have more expressive options to describe results in general and set-theoretic topology, but perhaps even support results beyond those which can be proven in ZFC?

As an easy example: does not currently appear in a search for Cardinality $\leq \aleph_1$ + Separable. But it could if we had an [x] Assume CH option that tells Cardinality to assume that new Cardinality("aleph 1") == new Cardinality("beth 1").

Questions? Answers? Volunteers?

References

• C.; Williams, Marshall. "Separation Axioms Among US." Zbl 08109626. Topology Appl. 375, Article ID 109467, 11 p. (2025).
• C. "Database-Driven Mathematical Inquiry and the -Base Model for Small Semantic Databases." LuCaNT: Databases, Algorithms, and Computational Number Theory. Contemporary Mathematics. American Mathematical Society. (2026?).
• C. "Non-Hausdorff Properties." Colloquium Mathematicum. (2026?).