A Space Without a Single Point

In 1607, in Beijing, the scholar Xu Guangqi and the Italian Jesuit Matteo Ricci published a Chinese translation of the first six books of Euclid’s Elements. The first line of the first book is just four characters long: 点者无分.

For more than two thousand years, geometry has started from this sentence. First come points; points make lines, lines bound surfaces, surfaces enclose solids. At the end of the nineteenth century, when Georg Cantor created set theory, the picture hardened into a piece of modern common sense that almost nobody questions:

A space is a set of points, plus a little structure.

The plane is a set of points. So is the sphere. So is the three-dimensional space we live in.

And yet in the twentieth century a group of mathematicians did something that sounds almost absurd. They built a space without a single point.

This is not the empty set. The empty set has nothing in it. This space has regions—big ones and small ones, regions that overlap and regions that sit inside other regions—and part of it can even be “everywhere.” You just can’t find a point in it.

More surprising still, this is not a word game. Follow the road far enough and you arrive at the topos, which Alexander Grothendieck would later call the vastest thing he ever conceived. Along the way you get a new understanding of the key Paul Cohen used to show that the continuum hypothesis cannot be proved, and you even walk into one of the most puzzling theorems of quantum mechanics.

So: how do you take the points out of a space? What is left once they’re gone? And why does it matter?

1You have never seen a point

Try a small experiment. Someone asks, “Where are you right now?”

You might say, “In the United States.” Too vague. “In New York.” “In Manhattan.” “On the red steps in Times Square.” You can keep going, each answer sharper than the last. But every answer you give is a region, not a point. Even the blue dot on your phone’s map sits inside a pale circle of uncertainty—and that circle is what the phone actually knows.

Physical measurement works the same way. Measure a table with a ruler and you get “1.20 meters, give or take 5 millimeters”: the length lies somewhere in an interval. A better instrument shrinks the interval, but a finite number of measurements only ever gives you a region.

So what is a point? It is the limit you imagine when the measuring goes on forever and the regions shrink without end—an idealization.

Around 1920, the philosopher and mathematician Alfred North Whitehead (the Whitehead who wrote Principia Mathematica with Bertrand Russell) argued in earnest that geometry should start from regions rather than points. A point, on his account, is nothing but what a nested sequence of ever-smaller regions “points to.”

The branch of mathematics that studies the shape of spaces is called topology, and, tellingly, its basic unit is not the point. It is a kind of region called an open set. Theoretical computer scientists (Smyth, Abramsky, Vickers and others) gave open sets a beautiful interpretation:

An open set is a property that can be confirmed by a single, finite observation.

Take “the table is longer than 1.19 meters.” If your measurement reads 1.20 ± 0.005 m, the whole error bar lies above 1.19, so the property is confirmed. And if the table were a hair longer or shorter, the verdict wouldn’t change: there’s wiggle room. A region with wiggle room is an open set.

“The table is exactly 1.2 meters long,” on the other hand, is not open. However precise your instrument, the error bar has some width, and nothing with width fits inside a point with none. You can never confirm it.

Read this way, the two slightly odd rules in every topology textbook suddenly look inevitable:

• A union of any number of open sets is open. To confirm “A₁ or A₂ or A₃ or …” (even infinitely many), it is enough to confirm one of them—a finite amount of work.
• Only an intersection of finitely many open sets is guaranteed to be open. To confirm “A₁ and A₂ and A₃ and …” infinitely many times over, you would need infinitely many experiments, which no finite observation can deliver.

In other words, topology is at heart a logic of observable properties. What it really cares about is how regions relate to one another; points just hide in the background, as limits.

Which raises a bold question: what if we dropped the points altogether and kept only the regions?

2Taking the points away

Imagine a map with no coordinates, no place names, no “locations” of any kind—only regions and the relations between them:

• two regions have an overlap (“and”);
• any number of regions can be merged into one (“or”);
• there is a largest region (“everything”) and a smallest one (“nothing”);
• these operations obey a few natural laws, such as distributivity: the overlap of A with “B₁ or B₂ or …” is “A overlap B₁” or “A overlap B₂” or ….

That’s all. Nowhere do the rules mention a point, and nowhere do they say what regions are made of.

Mathematicians such as Charles Ehresmann began studying structures like this in the 1950s. In 1972 John Isbell gave them a name—locales, literally “places”—and the subject acquired a name that sounds like a joke: pointless topology. Peter Johnstone could not resist the pun: his celebrated 1983 survey is titled The point of pointless topology.

So the points are gone. Still, we’d like to know: what does “a point” mean in this new setting?

The answer is neat:

A point is an answer sheet that responds to every region, answering “in” or “not in.”

The answers can’t be arbitrary. They have to be consistent:

1. Answer “in” for everything and “not in” for nothing;
2. You are in A and in B exactly when you are in the overlap of A and B;
3. If you are in a region made by merging other regions, you must be in at least one of them.

Rule 3 is the crucial one: a point can’t be smeared out. It can’t claim to be “in the north half or the south half of the city” without being in either. It has to actually land somewhere.

Try it on the real number line. The answer sheet of √2 begins like this:

This infinitely long answer sheet pins down √2 completely. And for the spaces we know well—the real line, the plane, the sphere—one can prove that every point produces a valid answer sheet, and every valid answer sheet comes from exactly one point. Nothing is lost.

So pointless topology doesn’t throw ordinary topology away. It contains all of it, written differently: regions come first, and points are read off from the regions.

But there’s a loophole. Nothing in the rules guarantees that a valid answer sheet exists.

So you could have something with plenty of distinct regions—overlapping, nested, richly structured, certainly not “empty”—and not a single valid answer sheet.

That is a space without a single point.

Does such a thing actually exist? It does. It’s hiding in the most familiar space of all: the real line.

3Where the rationals meet the irrationals

In school we learn that real numbers come in two kinds: rational numbers, which can be written as fractions (like 1/3 or 0.25), and irrational numbers, which can’t (like √2 or π).

They have one remarkable thing in common: both are everywhere. Take any stretch of the number line, however short—a billionth of a unit, say—and it contains both rationals and irrationals. Mathematicians call this being “dense.”

On the other hand, they share no points at all. No number is both rational and irrational. Two things that are everywhere, with an empty intersection.

This quietly offends a topologist’s intuition. If two things are both all over the place, shouldn’t what they have in common be all over the place too?

Now move the rationals and the irrationals into the pointless world. There, a subspace isn’t recorded by which points it contains but by how it sees each region: the rationals see the rational part of every region, the irrationals see the irrational part. Neither ever mistakes a nonempty interval for an empty one.

In 1972 Isbell proved a theorem:

In the pointless world, the intersection of any number of dense subspaces is still dense.

The rationals are dense; so are the irrationals. So their “intersection” in the pointless world is not empty at all—it is spread densely across the entire real line. Look through any open interval, however small, and you will see it.

So which points does it contain? A point in this intersection would have to lie in the rationals and in the irrationals at once. There is no such number.

So it contains no points at all.

Think of it as a ghost. Aim a microscope at any stretch of the real line and there is “something” in the viewfinder; reach in with tweezers to pick up a point, and there is nothing to pick up.

“Nonempty, yet without points” sounds like a contradiction, but the contradiction lives only in old habits of thought. In the pointless world, being nonempty is a property of a region—a region either is the empty one or it isn’t—not a matter of finding a point inside it. At no step does the argument use a point.

The ghost teaches us something: the pointless world restores the lovely intuition that two things which are everywhere should meet everywhere—at the price that where they meet is woven from pure region, with no points in it.

4Points were the troublemakers all along

You may be thinking: lovely, but what is it good for?

Here is one payoff that matters a great deal to mathematicians. Topology has a load-bearing result called Tychonoff’s theorem: take any number of compact spaces (roughly, spaces like the closed interval [0, 1] or a sphere—bounded, with nothing missing), multiply them together, and the result is still compact. A great deal of modern analysis rests on it.

But it has a dark secret. In 1950 John Kelley proved that Tychonoff’s theorem is equivalent to the axiom of choice—the axiom that lets you pick one item from each of infinitely many boxes all at once, and that also produces oddities like the Banach–Tarski paradox (cut a ball into finitely many pieces and reassemble them into two balls, each identical to the original). If you want Tychonoff’s theorem, you have to accept the axiom of choice.

In the pointless world, however, Johnstone proved in 1981 that Tychonoff’s theorem needs no axiom of choice at all.

Where did the axiom of choice go? In the classical proof, its whole job, in the end, is finding points—making sure a space has enough of them. The structure of the regions was fine all along; the infinite picking was needed only for the point-finding step. Take the points away, and the trouble leaves with them.

This is also why constructive mathematicians and theoretical computer scientists are so fond of pointless topology. Regions correspond to finite information, which can be computed and checked; points are often infinite objects—the very source of non-constructiveness.

5Grothendieck: replacing the regions too

So far we have only removed the points; the regions are still there. The next person in our story went further. He replaced the regions as well.

That person is Alexander Grothendieck, one of the most legendary mathematicians of the twentieth century—a man who walked away from mathematics at the height of his powers and spent his last decades in near-total seclusion in the Pyrenees.

The story starts with the Weil conjectures. In 1949 André Weil posed a set of conjectures about how many solutions polynomial equations have in finite number systems. A finite number system is something like clock arithmetic modulo 7: only the numbers 0 through 6, wrapping around like a clock, so that 6 + 1 comes back to 0. Count the solutions of polynomial equations in such worlds, Weil guessed, and the counts follow astonishingly regular patterns—the deepest of which is an analogue of the famous Riemann hypothesis.

Weil also saw something else: if these algebraic objects could be given a “cohomology theory” of the kind topologists use—a tool for precisely counting a space’s holes—the conjectures might come within reach.

The trouble was that the natural open sets on these algebraic objects (the so-called Zariski topology) are far too coarse: in the most common cases, any two nonempty open sets must intersect. Probing for holes with open sets like these is like fishing with a net whose mesh is bigger than a football field. You catch nothing.

Grothendieck’s idea was revolutionary: who says an “open set” has to be a subset?

Picture a spiral staircase. Seen from directly above, its shadow on the ground is a circle. Every short piece of the staircase sits exactly over a short arc of the circle and looks just like it; but the staircase as a whole is clearly not part of the circle—it winds around again and again.

Something that looks locally exactly like a small piece of the original space, but is globally something else, is called an étale map. Grothendieck’s move was: fine, let such things serve as open sets too. What matters is not whether an open set is a subset, but which families of things count as covering a region. Once the rules of covering are spelled out, the whole machinery of topology runs as before. This generalized notion is now called a Grothendieck topology.

With the new “open sets” in place, one more ingredient is needed: the sheaf.

A sheaf, put simply, is local data that can be glued together. Think of an atlas: each page maps one region, neighboring pages agree where they overlap, and so the local maps glue together—in exactly one way—into a map of the whole. Or think of temperatures across a country: each small area has its own temperature map, and as long as the maps agree where they overlap, they assemble into one map of the whole country.

Local data + agreement on overlaps → a unique global whole. That is the entire spirit of a sheaf.

Sometimes, though, everything makes sense locally and still refuses to fit together globally.

You have surely seen the Penrose staircase—the one M. C. Escher used in his print Ascending and Descending. Walk along it and every step takes you up; go all the way around and you are back where you started. Cover up most of the picture, and any small piece you look at is a perfectly ordinary staircase.

In the language of sheaves: on each short piece of the staircase we can assign heights to the steps, and we can make each piece agree with the next—but come all the way around, and the last piece refuses to agree with the first. Locally solvable, globally unsolvable.

This kind of obstruction—solvable locally, unsolvable globally—can be measured precisely. It is called cohomology. In 1991 Roger Penrose himself wrote a paper titled On the cohomology of impossible figures.

Back to Grothendieck. He realized that when you study a space, what really matters is neither its points nor its open sets but all the sheaves on it, together with the relationships among them. All the sheaves on a space form a vast mathematical world—and that world is a topos. “Topos” is Greek for “place”; fittingly, so is “locale.”

For Grothendieck, the topos itself was the true space. Ordinary spaces can be turned into toposes without losing any information (at least for well-behaved spaces), but many toposes come from no ordinary space at all. In the famous 1963–64 seminar known as SGA 4, Grothendieck and his collaborators built up topos theory and étale cohomology systematically; in 1974 his student Pierre Deligne proved the hardest part of the Weil conjectures.

What, then, is a “point” of a topos? Roughly, it is a way of standing somewhere: from that spot you can read off a piece of local data from every sheaf (mathematicians call it a stalk), in a way compatible with every construction.

Deligne proved that a large class of well-behaved toposes always have enough points. But a general topos can have none at all. The rational-meets-irrational ghost from earlier is an example: the sheaves on it form a topos with no points whatsoever. In Grothendieck’s sense, it is still a perfectly genuine space.

6From outside, a space; from inside, a universe

Around 1969–70, F. William Lawvere and Myles Tierney made another astonishing discovery: Grothendieck’s toposes can be characterized by just a handful of axioms—and those axioms describe precisely a world in which you can do mathematics, the way you do in set theory.

Inside a topos you can talk about objects, functions, and subsets; you can build products, function spaces, the natural numbers… Most everyday mathematical constructions go through. A topos is a whole mathematical universe.

But every universe has its own laws of physics—starting with logic.

In the familiar universe of sets, every statement is either true or false: there are two truth values. In the universe of sheaves on a space, the truth value of a statement is a region: where it holds.

Here is an example. On a weather map, the truth value of “it’s raining here” is the rainy region. And the truth value of “it’s not raining here”? It is the region where you can confirm that it isn’t raining: the open region outside the rain.

So what about “it’s raining or it isn’t”? Common sense says this holds everywhere. But in this universe its truth value is the union of the two regions—which misses exactly the boundary of the rain. Stand on the boundary and, however small the area you observe, it always contains both rain and no rain: you can neither assert “it’s raining” nor assert “it’s not.”

In other words, the law of excluded middle fails here, and it fails exactly on the boundary. Logic without excluded middle is called intuitionistic logic. It grew out of the “intuitionism” launched by the Dutch mathematician L. E. J. Brouwer in the early twentieth century, for philosophical reasons—and topos theory shows that it is simply the logic geometric spaces come with.

Different toposes are universes with different laws. For example:

• In the “effective topos” that Martin Hyland built in 1982, every function from natural numbers to natural numbers is computable;
• In a “smooth topos” (synthetic differential geometry) there are infinitesimals whose square is zero—not all of them are zero, yet you can never exhibit one that is definitely nonzero—and every function from the reals to the reals is smooth. The infinitesimals that the founders of calculus happily threw away once squared finally get a rigorous home;
• In some toposes, the continuum hypothesis is false—which is the story of the next section.

These are the two faces of a topos:

From outside, it is a (generalized) space; from inside, it is a mathematical universe.

That is exactly why toposes became a bridge between geometry and logic. In his memoir Récoltes et Semailles (“Harvests and Sowings”), Grothendieck likened the topos to the bed of a deep river, where geometry and algebra, topology and arithmetic, mathematical logic and category theory, the continuous and the discrete all flow together. Today the mathematician Olivia Caramello pushes in this direction, using toposes as “bridges” between theories: the same topos often has several very different presentations, each corresponding to a different mathematical theory, so a result in one field can be carried across the topos into another that looks entirely unrelated.

7A space full of impossible things

Now we can tell the most magical part of the story.

Georg Cantor proved, with his famous diagonal argument, that the real numbers cannot be listed. However you write down r₁, r₂, r₃, …, some real number is left out.

So consider this “space”: all the ways of listing every real number.

By Cantor’s theorem, this space has no points, because no such list exists. In the world of points, the story ends there: an empty set, nothing more to say.

But in the pointless world we don’t look at points; we look at regions. The basic regions of this space are finite descriptions of a list, such as “the 1st entry is π, the 5th entry is √2.”

Each such description carves out all the lists that satisfy it—a region. The key fact is that finitely many descriptions that don’t conflict can never lead to a contradiction: whichever positions you have filled, infinitely many are still free; whichever real number you want to include, there is always room for it. Cantor’s contradiction erupts only at the moment you demand that every position be filled and every real number appear.

So we get a locale with infinitely many nonempty regions and a rich structure—and not a single point, because a point would be a complete answer sheet: a genuine, complete list.

It is a space full of impossible things.

Now for the magical step. Don’t look at it from outside; move in—into the universe formed by the sheaves on this space. There you will find that such a list really exists. The old real numbers have become listable in the new universe.

Doesn’t that contradict Cantor? No. In the new universe, “all the real numbers” is a bigger collection, and the old real numbers are just part of it. Cantor’s theorem still holds inside the new universe—what got listed is simply no longer all of the new universe’s real numbers.

Readers who know some logic may recognize this: it is the core idea of forcing. In 1963 Paul Cohen invented forcing to prove that the continuum hypothesis—“there is no infinite size strictly between the number of natural numbers and the number of real numbers”—cannot be derived from the standard axioms of mathematics. (Kurt Gödel had already shown, by 1940, that it can’t be refuted either.) Cohen won the Fields Medal for it in 1966. The essence of forcing is to add objects to the mathematical universe that weren’t there before—lots of brand-new real numbers, for instance. Set theorists call such objects generic.

Later, Lawvere and Tierney found that, translated into the language of toposes, Cohen’s construction amounts to this: take a suitable space with no points, and move into the universe of sheaves on it. From outside, that “point” can’t be found no matter how hard you look. Step inside, and there it is.

A space without points is exactly where things that don’t exist yet can hide.

8Bonus: the quantum Penrose staircase

To finish, a story from physics.

In classical mechanics, the state of a system is a point in its phase space: position, momentum, energy—every physical quantity has a definite value at that point.

Quantum mechanics is different. In 1967 Simon Kochen and Ernst Specker proved a famous theorem: for sufficiently complex quantum systems (a spin-1 particle, for instance), it is impossible to assign definite values to all physical quantities in advance while making the quantities that can be measured together satisfy the relations they must.

Yet if you look only at a set of mutually compatible quantities that can be measured together—standing at one “classical viewpoint,” so to speak—assigning definite values is no problem at all.

Every local viewpoint makes sense; they just won’t fit together globally. Sound familiar? It is a quantum Penrose staircase.

In 1998 the physicist Chris Isham and the philosopher of physics Jeremy Butterfield glued all the classical viewpoints together into a sheaf-like object, as a candidate for the quantum system’s “state space.” They proved that the Kochen–Specker theorem is exactly equivalent to the statement that this state space has no global points (in mathematical terms, no global sections). Since then Andreas Döring and Isham—and Chris Heunen, Klaas Landsman, Bas Spitters and others—have developed a research program that describes quantum theory with toposes.

Whether this is the right road to understanding the quantum world is far from settled among physicists. But at least in this picture, the state space of the quantum world is precisely a space without a single point.

CodaA point is our name for infinite precision

Back to the sentence we started with: a point is that which has no part.

More than two thousand years ago, Euclid set geometry off from the point; four centuries ago, that sentence arrived in China. Today we know that points are not the most basic building blocks of space. A point is more like a limit, an ideal—something we imagine once we push observation to infinite precision.

Most of the time that limit exists, and classical geometry is safe and sound. But when it doesn’t, the absence of points becomes the most interesting information of all:

• Sometimes it is a rescued intuition—two things that are everywhere meet everywhere;
• Sometimes it is an object not yet born—the new real numbers Cohen added to the universe;
• Sometimes it may be the way nature itself is—in the quantum world, there is no state in which every quantity has a definite value.

A space without a single point looks empty from outside, with nowhere to stand. Step inside, and there is a whole universe.

A point is our name for infinite precision. But the world only ever gives us finite precision.

Perhaps space was never made of points at all, but woven from regions.