Academic Research Paper
The Hidden
Intersections
How point-free topology reveals that the Banach-Tarski pieces were never truly disjoint — and the paradox dissolves.
The Banach-Tarski Paradox
A theorem so counterintuitive it earned the name “paradox” despite being a rigorous proof.
What the Theorem Says
A solid ball B³ can be decomposed into a finite number of pieces (five suffice) and reassembled using only rigid motions (rotations and translations) into two solid balls, each identical to the original.
- 1Take a solid ball in ℜ³
- 2Cut it into 5 disjoint pieces (as point-sets)
- 3Rotate and translate the pieces (no stretching)
- 4Obtain two complete copies of the original ball
Proved by Banach & Tarski (1924). Requires the Axiom of Choice.
Why It's Called a Paradox
Rigid motions preserve volume. One ball has volume V. Two balls have volume 2V. Something has gone terribly wrong:
- “Volume was created from nothing”
- “Conservation of measure is violated”
- “Physical reality can't work this way”
- “Mathematics must be broken somewhere”
But what if the pieces aren't really disjoint?
The Standard Responses
For a century, mathematicians have offered three ways to live with the paradox.
The pieces are non-measurable — volume is simply undefined for them. No contradiction because there's nothing to conserve.
Technically correct, but unsatisfying
Work in the Solovay model where all sets are measurable. But this costs Hahn-Banach, Tychonoff, and much of functional analysis.
Too expensive — cures the disease by killing the patient
Replace point-set topology with locale theory, where 'part of a space' means something richer than 'subset of points.' The pieces overlap.
Our approach — works in full ZFC
Two sets can share no points and still overlap as parts of a space
In locale theory, “part of a space” means something richer than “subset of points.” A sublocale captures how a part sits within the ambient topology. Two sublocales can be point-disjoint yet topologically intertwined.
A Concrete Example
Consider the rational numbers ℚ and the irrationals ℝ \ ℚ inside the real line.
As point-sets:
Completely disjoint. ℚ ∩ (ℝ \ ℚ) = ∅. Not a single point in common.
As sublocales:
Both are dense. Their meet contains S&sub0; (the regular opens). They share the entire topological skeleton of ℝ.
The Banach-Tarski pieces are exactly like this: point-disjoint but everywhere dense, filling every neighborhood of B³. As sublocales, they necessarily overlap.
Five Steps to Dissolution
Each step is independently verifiable. Together, they form a complete resolution of the Banach-Tarski paradox within full ZFC.
The Pieces Are Dense
Every Banach-Tarski piece meets every open ball in B³. The pieces are everywhere dense — they thread through the entire sphere.
Dense Subsets Become Dense Sublocales
The canonical embedding from subsets to sublocales preserves density. A dense subset of ℝⁿ maps to a dense sublocale.
Dense Sublocales Always Overlap
Isbell’s density theorem: the meet of any family of dense sublocales is dense. Dense sublocales cannot be disjoint.
The Overlap Has Full Measure
Simpson’s theorem: Lebesgue measure extends to all sublocales via outer measure. The shared sublocale S₀ has μ*(S₀) = μ(B³).
The Paradox Is Blocked
Finite additivity requires disjoint pieces. The BT pieces are not disjoint as sublocales. So you cannot add their measures.
BT vs. Dougherty-Foreman
The Dougherty-Foreman paradox uses open sets as pieces. Open sets are complemented in the locale — genuinely disjoint as sublocales. The locale framework correctly distinguishes the two cases.
| Property | Banach-Tarski | Dougherty-Foreman |
|---|---|---|
| Pieces are... | Non-measurable, everywhere dense | Open sets (Borel measurable) |
| Disjoint as sets? | Yes | Yes |
| Disjoint as sublocales? | No — dense sublocales always overlap | Yes — open sets are complemented in the frame |
| Uses Axiom of Choice? | Yes (non-constructive pieces) | No (explicit construction) |
| Paradox in locale framework? | No — additivity is blocked by overlap | Yes — genuinely disjoint, genuinely paradoxical |
Pieces are...
Non-measurable, everywhere dense
Open sets (Borel measurable)
Disjoint as sets?
Yes
Yes
Disjoint as sublocales?
No — dense sublocales always overlap
Yes — open sets are complemented in the frame
Uses Axiom of Choice?
Yes (non-constructive pieces)
No (explicit construction)
Paradox in locale framework?
No — additivity is blocked by overlap
Yes — genuinely disjoint, genuinely paradoxical
The locale framework doesn't blindly dissolve all paradoxes — it makes precise distinctions. BT is blocked because its pieces overlap as sublocales. DF goes through because its pieces are genuinely disjoint.
Why Locales Are the Right Setting
Locale theory isn't an ad hoc fix for one paradox. It's an independently motivated framework that resolves BT as a side effect.
Tychonoff Without Choice
In locale theory, the Tychonoff theorem holds without the Axiom of Choice. Products of compact locales are compact — no selection axiom needed.
Constructive Mathematics
Locales are the natural framework for topology in constructive and predicative mathematics. Results that require AC classically often become constructive in locale theory.
Physical Space
Physical space has no dimensionless, sizeless points. Locales model space via its observable structure — the open sets — without requiring a substrate of points.
Read the Full Paper
Selected References
Simpson, A.
Measure, randomness and sublocales
APAL 163(11), 1642–1659
Picado, J. & Pultr, A.
Frames and Locales: Topology without Points
Birkhäuser
Leroy, J.
Un théorème de Banach-Tarski dans le cadre localique
arXiv:1303.5631
Lehner, T.
Measure theory via locales
arXiv:2510.08826
Isbell, J.
Atomless parts of spaces
Math. Scand. 31, 5–32
Wagon, S.
The Banach-Tarski Paradox
Cambridge University Press
Dougherty, R. & Foreman, M.
Banach-Tarski decompositions using sets with the property of Baire
JAMS 7(1), 75–124
Johnstone, P.T.
Stone Spaces
Cambridge University Press
“The Banach-Tarski pieces are not disjoint parts of the sphere. They are dense, overlapping aspects of it — and the paradox was always an artifact of confusing points with parts.”