Closed sets obey the continuum hypothesis

The continuum hypothesis says that every set of real numbers is either countable or as large as the whole line — that no size sits strictly between the whole numbers and the points of a line. The size that cannot be pinned down explained that the usual axioms of set theory neither prove nor refute it. What counting can prove exists showed why arbitrary sets of reals are hard to say anything about: most of them are not described by any countable amount of information.

But the sets that mathematics actually uses are. And for the simplest of them, the closed sets, the continuum hypothesis is simply true, by an argument Georg Cantor and Ivar Bendixson completed in 1883 — before anybody knew the general question was unanswerable.

The picture is the tool. The set in its top row consists of the numbers $\tfrac1a + \tfrac1b$ with $a, b \ge 2$, the numbers $\tfrac1a$, and $0$. It is closed: every point it approaches, it contains. Each row below is obtained from the one above by deleting every isolated point — every point with a small neighbourhood containing no other point of the set — and after three such deletions nothing is left.

Taking away what stands alone #

A point of a set is isolated if some small interval around it contains no other point of the set, and it is a limit point if every interval around it, however small, contains others. Cantor called the set of limit points the derived set, and deleting the isolated points of a closed set is the same as passing to its derived set.

In the top row, a sum $\tfrac1a + \tfrac1b$ is isolated: the other sums of the same form keep a definite distance from it. But $\tfrac1a$ is a limit point, because $\tfrac1a + \tfrac1b$ comes arbitrarily close to it as $b$ grows. And $0$ is a limit point of the $\tfrac1a$’s. So the first derived set is $\{\tfrac1a\} \cup \{0\}$, the second is $\{0\}$, and the third is empty. The number of steps it takes to empty the set measures how deeply its points are nested inside one another, and that number is its Cantor–Bendixson rank.

Sums of up to three reciprocals take four steps; sums of up to $d$ take $d + 1$. And the process need not stop at a finite number. Place, near each point $\tfrac1k$, a small copy of a set of rank $k$, shrinking the copies so that they crowd onto $0$: every finite number of deletions leaves some of them, but the intersection of all the finite stages — the set left after infinitely many deletions — is just $\{0\}$, and one more deletion empties it. That set has rank $\omega + 1$, the first infinite ordinal and one more.

Cantor invented the transfinite ordinals for exactly this purpose. The sequence of derived sets needed indices beyond the whole numbers — the derived set after all finite stages, then its derived set, and so on — and the ordinals of one step in front of infinitely many were first written down, in 1880, as labels for the stages of this process.

Why the process stops at a countable stage #

For a countable closed set, the deletions empty it at some countable stage — that is, before the first uncountable ordinal $\omega_1$. For a general closed set they may never empty it, but they always stop, reaching a set that the next deletion leaves unchanged, and they stop at a countable stage too.

The reason is a counting argument of the kind the previous tiers used. Each point deleted at some stage was isolated at that stage, so there is an interval with fractional endpoints containing it and no other point of the set at that stage. Assign each deleted point such an interval. Two points deleted at different stages cannot get the same interval — the later one was still present when the earlier one was deleted, and the earlier one’s interval contains no other point — and two deleted at the same stage cannot either. So distinct deleted points get distinct intervals with fractional endpoints, of which there are only countably many. Only countably many points are ever deleted, and a process that deletes at least one point at every stage until it stops can run through only countably many stages.

What the process leaves #

When the deletions stop, what is left is a closed set with no isolated points at all. Such a set is called perfect, and the whole interval, the middle-thirds set and the empty set are examples.

A non-empty perfect set is as large as the line, and the proof is the figure. Because no point is isolated, any closed piece that contains a point of the set in its interior contains other points of the set too — so two disjoint closed pieces can be found inside it, each still containing points of the set. Split each of those in two, and so on, shrinking the pieces as the levels go down. Every infinite sequence of left and right choices follows a nested sequence of pieces, the pieces close down on a single point, and different sequences reach different points. So the perfect set contains a copy of the middle-thirds set, and has at least as many points as there are infinite sequences of two choices — which is the size of the line.

The theorem #

Put the two halves together.

The Cantor–Bendixson theorem. Every closed set of reals is the union of a perfect set and a countable set, with nothing in common between them.

The perfect part is what the deletions leave when they stop; the countable part is everything they delete. In the figure the perfect part is the middle-thirds set, which no deletion touches, and the countable part is the reciprocal sums, which three deletions remove. The theorem says that every closed set looks like this: a perfect kernel, possibly empty, with countably many points scattered around it.

And the continuum hypothesis for closed sets follows in one line. If the perfect part is empty, the set is countable. If it is not, the set contains a perfect set, which is as large as the line, so the set is as large as the line. No closed set has a size strictly between. Whatever the answer to the continuum hypothesis for sets in general, no counterexample can be closed.

Two numbers classify every countable closed set #

The rank does more than measure the length of a process. Together with one other number, it says exactly what a countable closed set looks like.

Take a countable closed and bounded set and run the deletions. The stage just before the set becomes empty holds finitely many points — an infinite set there would have a limit point, by boundedness, and so would survive one more deletion. Call the number of steps the rank and the number of points at that last non-empty stage the degree. Stefan Mazurkiewicz and Wacław Sierpiński proved in 1920 that these two numbers determine the set completely up to a continuous deformation with a continuous inverse: two countable closed bounded sets with the same rank and degree can be carried onto each other point for point, continuously both ways, and two with different ones cannot.

They also found a standard example of each. The ordinals themselves, arranged in order with their natural notion of closeness — a limit ordinal is the limit of the ordinals below it — form such sets: the ordinals up to and including $\omega^\alpha \cdot n$ make a countable closed set of rank $\alpha + 1$ and degree $n$. So the set in the first figure, of rank three and degree one, is a disguised copy of the ordinals up to $\omega^2$, and every countable closed set is a disguised copy of some ordinal. The ordinals that are reached from below are, as spaces, the whole zoo of countable closed sets.

A theorem from Fourier series #

The problem that led Cantor to derived sets was not about sizes at all, and the connection is one of the more surprising origins of a mathematical subject.

In 1870 Cantor proved that if a trigonometric series — a sum of sines and cosines like the ones that build a square wave — converges to nought at every point, all its coefficients are nought: a function has at most one such expansion. He then asked how far “every point” could be weakened. If the series converges to nought except on a finite set, the coefficients are still nought. Except on a set with finitely many limit points — still nought. Except on a set whose derived set, taken again and again, becomes empty after finitely many steps — still nought.

To state the theorem he needed the derived sets and their iteration, and to go past finitely many iterations he needed the stages after all finite ones. The transfinite ordinals, the theory of infinite sizes and in the end the continuum hypothesis itself all grew out of a question about when a trigonometric series is determined by its sum. The sets of points that can be ignored without losing uniqueness are still studied under the name sets of uniqueness, and deciding which closed sets they are turned out to be a problem of exactly the descriptive-set-theoretic kind this essay is about.

Up the hierarchy of definable sets #

The closed sets are the simplest level of a hierarchy of increasingly complicated definable sets, and the natural question is how far the answer extends.

The property that did the work for closed sets is that every uncountable member contains a perfect set — the perfect set property. Pavel Alexandrov and Felix Hausdorff proved it independently in 1916 for the Borel sets, which are built from the closed ones by countable unions and complements. Mikhail Suslin proved it in 1917 for the analytic sets, the shadows of Borel sets that counting showed need not be Borel themselves. For all of these the continuum hypothesis holds: an uncountable Borel or analytic set is as large as the line.

One step further up, the proofs stop. The co-analytic sets are the complements of analytic ones, and Kurt Gödel observed in 1938 that in his constructible universe — the model in which he proved the continuum hypothesis consistent — there is an uncountable co-analytic set containing no perfect set. So the usual axioms cannot prove the perfect set property for co-analytic sets, by the same method two worlds that both obey the rules describes: a world satisfying every axiom in which the statement fails. Nor can they refute it: Robert Solovay showed in 1970 that, assuming a large cardinal is consistent, so is the statement that every set definable from real numbers and ordinals has the perfect set property. With stronger large-cardinal axioms the property holds for every set in the whole projective hierarchy. Where the definitions get complicated enough, the axioms stop deciding.

For arbitrary sets the perfect set property is simply false, if the axiom of choice holds. Felix Bernstein built in 1908 an uncountable set that neither contains nor misses any perfect set entirely — it meets every perfect set and every perfect set meets its complement — by listing the perfect sets, of which there are as many as points, and choosing a point for the set and a point for its complement from each in turn. A Bernstein set has the size of the line, so it is not a counterexample to the continuum hypothesis; it only shows that the property which proved the hypothesis for closed sets cannot prove it for all.

Beyond the line #

Nothing in the argument used the real line specifically. The deletions, the counting of intervals with fractional endpoints and the splitting of perfect sets all work in any space with a countable collection of basic open sets and a notion of convergence in which nested closed pieces shrinking to nothing meet in a point — a Polish space, in the language of the subject. The space of infinite sequences of noughts and ones, the space of continuous functions on an interval, the space of all countable graphs on the whole numbers coded as sequences: in each, every closed set is countable or contains a copy of the middle-thirds set, and so has the size of the line.

That generality is what makes the theorem useful rather than curious. Questions about how many objects of some kind there are — countable models of a theory, isomorphism types of countable structures, orbits of a group action — become questions about definable sets in one of these spaces, and the perfect set property for the relevant class answers them. Vaught’s question below is the most famous case in which the answer has not yet been found.

What the figures show of an infinite process #

Every set drawn is truncated. The reciprocal sums are drawn with denominators up to forty or fourteen, the splitting to five levels and the middle-thirds set to its fifth stage. The claims about isolated points and limit points are claims about the infinite sets, which are defined exactly by their formulas; the figures check them on the truncations — every point claimed to be a limit has points of the next kind crowding onto it within the resolution the truncation allows.

The transfinite stages are described and not drawn. A set of rank $\omega + 1$ is built in the text; no figure can display infinitely many deletions followed by one more.

The hierarchy table records theorems and independence results, not computations. Nothing in a figure can show that a statement is unprovable; that is a theorem about the axioms, proved by building models in which the statement fails.

Still open: the same dichotomy for countable models #

The perfect set property is a dichotomy: a definable set is either countable or contains a copy of the middle-thirds set. The most famous open question of this shape is not about sets of reals but about models.

Take a first-order theory in a countable language — the axioms of groups, of orders, of any structure — and count its countable models, identifying models that are isomorphic. Robert Vaught conjectured in 1961 that the number is either countable or exactly the size of the line: no complete theory has exactly $\aleph_1$ countable models, if $\aleph_1$ is less than the size of the line. It is the continuum hypothesis asked of isomorphism classes rather than of sets, and the tools that settled it for closed and analytic sets settle part of it: Michael Morley proved in 1970 that the number is at most $\aleph_1$ or exactly the size of the line, and John Burgess and Jack Silver extended the dichotomy to wide classes of definable equivalence relations. Vaught’s conjecture itself has been proved for many special kinds of theory and remains open in general.

A question answered where the sets are simple #

The habit worth keeping is the direction the answer came from.

The continuum hypothesis is undecidable for sets in general, and it is tempting to conclude that it is a question without content. The Cantor–Bendixson theorem shows otherwise: for the sets anyone meets in analysis, the answer is yes, by an argument that strips a closed set down, one layer of isolated points at a time, to a perfect kernel whose size is forced. The question has an answer exactly as far as the sets can be described, up to the analytic sets by proof, a little further under large-cardinal axioms, and not at all for sets built by arbitrary choices. The undecidability lives entirely among sets that nobody can write down. That is a statement about where the difficulty is, not a dismissal of it: the sets that are hard to describe are most of the sets there are, and the continuum hypothesis is about all of them.