There are spaces with no points at all – placing a sixth root for the word point.
Paper 148 listed five distinct roots for the word point and wrote that the separator is which test object one maps in from. This paper adds a sixth, and that sixth can have no instances at all. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Frames and locales, sobriety, Heyting algebras, and the pointlessness of atomless complete Boolean algebras (Isbell, Johnstone) are all standard. It is not claimed that spaces are unreal, nor that points are a fiction.
First, the algebra of open sets does not satisfy excluded middle. In the Sierpinski space, whose opens are the empty set, a single-point set and the whole, take that single-point set. The interior of its complement is empty, and the interior of the complement of that is the whole space, so double negation does not return the original (computed here). The same happens on the real line: taking the union of the open interval from zero to one with the open interval from one to two, the interior of the complement is the part below zero together with the part above two, and the interior of the complement of that is the open interval from zero to two, so the point one comes back. The algebra of opens is a Heyting algebra and not a Boolean one, and the difference between double negation and the identity is exactly the point one had just removed.
Second, and this is the core. Points cannot be recovered from the open sets. For finite frames, every frame homomorphism into the two-element frame – that is, every map preserving bottom, top, finite meets and all joins – was enumerated. The two-point indiscrete space has two points while its locale has one. A three-point space with two points glued has three against two. By contrast the Sierpinski space gives two against two, and the two-point and three-point discrete spaces agree as well (computed here). The frame of the two-point indiscrete space is the two-element frame, which is precisely the frame of a one-point space, so no amount of looking at open sets can tell those two spaces apart. A space is a set of points with a topology has already failed here: what the opens remember is only the points that are topologically distinguishable.
Third, if the space is sober the points come back. For finite spaces, separating points and being sober coincide, and the counts in the table agree with that. The separator is therefore not space against locale but sobriety, which is the name of the condition under which points are recoverable. Since the real line and every Hausdorff space are sober, nothing is lost in ordinary spaces.
Fourth, there are locales with no points at all. The points of a complete Boolean algebra correspond bijectively to its atoms. But the measure algebra of the unit interval, the Borel sets modulo the null sets, is atomless: any element of positive measure splits into a proper part of positive measure, and bisecting repeatedly gives one half, one eighth, one thousand and twenty-fourth, and at twenty steps one part in one million forty-eight thousand five hundred and seventy-six, always positive, never reaching a least non-zero element (checked here). With no atoms there are no points. And the locale is far from trivial: uncountably many elements, a rich lattice structure, and a measure sitting on it. There is simply nothing that could be called a point. The space of points is empty while the space itself is not, and this is not a pathological construction but the algebra probability theory uses every day.
Fifth, a sixth root for the word point is thereby placed. A point of the topological space is a prime ideal, a closed point a maximal ideal, the generic point the zero ideal, a geometric point a map from an algebraically closed field, and a fat point a map from the ring with a square-zero element – so far Paper 148. The sixth is a point of a locale, a frame homomorphism to the two-element frame. And only the sixth has this property: a non-empty scheme always has points, since a non-zero ring has a prime ideal, whereas a non-empty locale may have none at all. Of the six roots of one word, only this one can come out empty. Paper 148 wrote that the separator is which test object one maps in from; here the test object is the two-element frame, and for some spaces there is not a single such map. The word point has a case with no instances.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / that spaces are unreal (not claimed) / that points are a fiction (not claimed). What is added: exhibiting the counts by enumerating every frame homomorphism on finite frames, writing that the separator is sobriety, and extending Paper 148’s census with a rung on which there may be no instances. What remains: thinking of a space as a collection of points is a habit formed among sober spaces. Carry only the open sets and points may or may not be recoverable, and sometimes there are none – and nothing is lost when there are none. One can still measure, still follow inclusions, still take unions and intersections. A point was never the material a space is made of, but a result that can sometimes be extracted from it.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). There Are Spaces With No Points at All [Paper 155]. Zenodo.
DOI: 10.5281/zenodo.22084848
ORCID: 0009-0006-8587-3450