Open Problems and the Research Frontier

Lindemann-Weierstrass theorem. If $\alpha_1, \dots, \alpha_n$ are distinct algebraic numbers, then $e^{\alpha_1}, \dots, e^{\alpha_n}$ are linearly independent over the field $\overline{\mathbb{Q}}$ of algebraic numbers.

Equivalently: if $\alpha_1, \dots, \alpha_n$ are algebraic numbers that are linearly independent over $\mathbb{Q}$, then $e^{\alpha_1}, \dots, e^{\alpha_n}$ are algebraically independent over $\mathbb{Q}$.

Status: settled, and awaiting an owner decision here. The theorem is a theorem: Lindemann proved the case that yields the transcendence of $\pi$ in 1882, and Weierstrass proved the general form in 1885. It is not deferred for a missing prerequisite in the sense of the rest of this category. It is reachable in principle from material this library is built to contain, but the development is long, and whether to author it has been flagged for a decision rather than answered. Until that decision is taken it is recorded here and used nowhere.

Not proved in this library. No page of this library proves the Lindemann-Weierstrass theorem, and nothing here may cite it as an established result. This item exists so that the transcendence facts about $\pi$ can be stated honestly rather than assumed.

What is known, and what would settle its place here. The proof is a quantitative refinement of Hermite's 1873 argument for the transcendence of $e$: one builds an auxiliary integral against a high power of a polynomial with the $\alpha_i$ as roots, uses the fundamental theorem of symmetric polynomials to show that the resulting algebraic sum is a nonzero rational integer, and then contradicts that with an analytic bound that forces it below $1$ in absolute value. The prerequisites are algebraic numbers and algebraic integers, symmetric polynomials, and elementary estimates on the exponential, all of which sit inside the intended scope of this library. What would settle its place is therefore a scope decision, not new mathematics: either the track is authored and this item is replaced by a proof-bearing theorem, or the decision is recorded to leave it out.

Why it matters here. Hermite's theorem that $e$ is transcendental is in scope and will be proved. Lindemann-Weierstrass is the next step up, and it is what delivers the transcendence of $\pi$, of $\sin 1$, of $\log \alpha$ for algebraic $\alpha \neq 0, 1$, and with them the impossibility of squaring the circle. Everything this library will be able to say about $\pi$ beyond irrationality is downstream of this one statement, so its status has to be recorded exactly rather than left vague.

Transcendence of $\pi$. The number $\pi$ is transcendental over $\mathbb{Q}$: it is not a root of any nonzero polynomial with rational coefficients. In particular $\pi$ is irrational, and $\pi$ is not constructible by straightedge and compass, so the circle cannot be squared.

Status: settled, and awaiting an owner decision here. Lindemann proved it in 1882. Like The Lindemann-Weierstrass theorem (awaiting a scope decision) ‡, of which it is the headline corollary, it is neither open nor blocked by a missing track; it has been flagged for a decision on whether to author the transcendence machinery, and until that decision is taken it is recorded and not used.

Not proved in this library. No page here proves that $\pi$ is transcendental, and no proof in this library may lean on it.

What is known, and what would settle its place here. The derivation from The Lindemann-Weierstrass theorem (awaiting a scope decision) ‡ is short: if $\pi$ were algebraic then so would $i\pi$ be, and $e^{i\pi} = -1$ together with $e^0 = 1$ would exhibit two exponentials of distinct algebraic numbers that are linearly dependent over $\overline{\mathbb{Q}}$, contradicting the theorem. So the whole cost sits in the theorem, plus Euler's identity, which this library does intend to prove once the complex exponential is built. Authoring both is what would replace this item by a theorem.

Why it matters here. Transcendence and irrationality are routinely confused, and the difference is exactly the difference between what this library proves and what it records. The irrationality of $\sqrt{2}$ is proved here, and the irrationality of $\pi$ alone has a famously short elementary proof (Niven, 1947) that is well inside scope. Transcendence is a strictly stronger and strictly more expensive statement, and the classical geometric consequence, the impossibility of squaring the circle, needs the stronger one. Recording the distinction is the point of this item.

Question. Is $e + \pi$ irrational?

Status: open. No proof and no disproof is known. The same is true of $e\pi$, of $e - \pi$, of $e/\pi$, of $\pi^{e}$, of $\pi^{\pi}$ and of $e^{e}$: for none of these is it known whether the number is rational, let alone whether it is transcendental. This is not a gap in this library's prerequisites. It is a gap in the subject.

Not proved in this library, and not provable anywhere at present. Nothing on any page here depends on the value or the arithmetic nature of $e + \pi$.

What is known, and what would settle it. Both constituents are settled individually: $e$ is transcendental (Hermite, 1873, a result that is in scope for this library and will be proved), and $\pi$ is transcendental (Transcendence of $\pi$ (awaiting a scope decision) ‡). A cheap symmetric-function argument already shows that the two candidates cannot both be tame: $e$ and $\pi$ are the roots of

$x^2 - (e + \pi)x + e\pi,$

so if $e + \pi$ and $e\pi$ were both algebraic then $e$ and $\pi$ would be algebraic too. Hence at least one of $e + \pi$ and $e\pi$ is transcendental, and that is essentially the whole of what is known about this pair. Note the asymmetry with $e^{\pi}$, which is known to be transcendental (Gelfond, via the Gelfond-Schneider theorem of 1934 applied to $e^{\pi} = (e^{i\pi})^{-i} = (-1)^{-i}$), and about which much more is known: Nesterenko (1996) proved that $\pi$ and $e^{\pi}$ are algebraically independent over $\mathbb{Q}$. By contrast $\pi^{e}$ is not known even to be irrational, because it is not of the form $\alpha^{\beta}$ with $\alpha, \beta$ algebraic and so Gelfond-Schneider says nothing about it.

Schanuel's conjecture, if proved, would settle all of these at once: it implies that $e$ and $\pi$ are algebraically independent over $\mathbb{Q}$, and hence that $e + \pi$ and $e\pi$ are both transcendental. Schanuel's conjecture is itself open, so this reduces one open problem to a much harder one rather than solving anything.

Why it matters here. The library proves irrationality where irrationality is provable, starting from the refutation of the claim that $\sqrt{2}$ is rational (FALSE: some rational number squares to 2 ↗), and it constructs $\mathbb{R}$ so that $e$ and $\pi$ can be defined at all. This item is the honest boundary marker: the elementary irrationality arguments do not scale, and the first genuinely simple combination of the library's two favourite constants is already past the edge of what anyone can prove.

Fix an integer base $b \ge 2$. A real number $x$ is normal in base $b$ if for every $k \ge 1$ each of the $b^k$ blocks of $k$ digits occurs in the base-$b$ expansion of $x$ with asymptotic frequency $b^{-k}$; it is absolutely normal if it is normal in every base $b \ge 2$.

Question. Is $\pi$ normal in base $10$? In any base?

Status: open. It is not known whether $\pi$ is normal in a single base. Much less is known than that: it is not even known whether every one of the digits $0, \dots, 9$ occurs infinitely often in the decimal expansion of $\pi$. The same questions are open for $e$, for $\sqrt{2}$, and for $\ln 2$. No naturally occurring constant has ever been proved normal.

Not proved in this library, and not provable anywhere at present. Nothing here depends on any digit statistic of $\pi$.

What is known, and what would settle it. Borel (1909) proved that almost every real number is absolutely normal, so normality is the typical behaviour and the exceptions form a null set; stating that theorem correctly needs the measure notions of the deferred measure and integration track. Explicit normal numbers are easy to write down once one stops asking for a familiar constant: Champernowne's constant $0.123456789101112\ldots$ is normal in base $10$, and Sierpinski and Turing gave constructions of absolutely normal numbers. Computations have checked hundreds of trillions of decimal digits of $\pi$ against the usual statistical tests, which they pass; the published record stood at $314$ trillion digits in November 2025, and the digit statistics of such runs are reported routinely. Passing a statistical test is evidence and not a proof, and no amount of computation can settle an asymptotic frequency. The most concrete programme is Bailey and Crandall's: the Bailey-Borwein-Plouffe formula for $\pi$ reduces base-$2$ normality of $\pi$ to a uniform-distribution statement about a specific chaotic iteration, their "Hypothesis A", which would settle the base-$2$ case if proved. Hypothesis A is open.

Why it matters here. This library defines $\pi$ analytically and will prove sharp facts about it, and a reader is entitled to ask what is not known. The answer is instructive: a constant can be pinned down exactly by a convergent series, be computable to arbitrary precision, and still resist the most basic question about its digits. Normality is also where analysis stops being able to help and measure-theoretic and number-theoretic tools take over, which is why it sits in this category rather than on a page.

For parameters $0 < a < 1$ and $b > 1$ with $ab > 1$, the classical Weierstrass function is

$W_{a,b}(x) = \sum_{n=0}^{\infty} a^{n} \cos(2\pi b^{n} x).$

It is continuous on $\mathbb{R}$ and, by Hardy's 1916 sharpening of Weierstrass's 1872 example, nowhere differentiable whenever $0 < a < 1$ and $ab \ge 1$. Its graph is a compact subset of $\mathbb{R}^2$, and the claim at issue is that its Hausdorff dimension is

$\dim_{H} \operatorname{graph}(W_{a,b}) = 2 + \log_{b} a = 2 + \frac{\log a}{\log b},$

a number strictly between $1$ and $2$, since $0 < a < 1$ gives $\log_b a < 0$ and $ab > 1$ gives $\log_b a > -1$.

Status: settled in 2018 for integer $b$, and open in general. Shen proved the formula for every integer $b \ge 2$ and every $a \in (1/b, 1)$. For non-integer $b$ the value of the Hausdorff dimension of the graph remains open. Even in the settled range the result is far out of reach here: it needs Hausdorff measure and dimension, hyperbolic dynamics, and the absolute continuity of an SRB measure on a solenoidal attractor.

Not proved in this library. Nothing here rests on the value of this dimension, and no page may cite the formula as established.

What is known, and what would settle the rest. The box-counting dimension of the graph is $2 + \log_b a$ and has been classical for decades (see Falconer, The Geometry of Fractal Sets); since Hausdorff dimension never exceeds box dimension, the entire difficulty is the lower bound. The successive advances were: Hunt (1998), who proved the formula for the randomly phased variant $\sum a^n \cos(2\pi(b^n x + \theta_n))$ for almost every phase sequence $(\theta_n)$; Barański, Bárány and Romanowska (2014), who proved it for integer $b$ and $a$ above a threshold $a_b \in (1/b, 1)$; and Shen (2018), who removed the threshold and covered every integer $b \ge 2$ and every $a \in (1/b, 1)$, by proving that the SRB measure of the associated solenoidal attractor is absolutely continuous. Ren and Shen (2021) then generalised the formula from $\cos$ to an arbitrary real analytic periodic $\varphi$, as a dichotomy: for integer $b \ge 2$ and $a \in (1/b, 1)$, either $\sum a^n \varphi(b^n x)$ is real analytic or its graph has Hausdorff dimension $2 + \log_b a$. Every one of these results keeps the hypothesis that $b$ is an integer. What would settle what remains is the non-integer case, where the associated dynamics is no longer a self-affine expanding map of a circle by an integer degree and the current method does not apply.

Why it matters here. Continuity and nowhere differentiability of a Weierstrass function are in scope for this library and will be proved from the Weierstrass M-test and a direct oscillation estimate. The dimension of the graph is not, and the gap between the two is worth naming: the elementary statement that the curve has no tangent anywhere is a nineteenth-century theorem, while the quantitative statement of how rough it is took until 2018 and is still incomplete. Recording that here keeps the nowhere-differentiability page from implying more than it proves.

Jacobian conjecture. Let $n \ge 1$ and let $F = (F_1, \dots, F_n) : \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial map whose Jacobian determinant

$\det\left(\frac{\partial F_i}{\partial x_j}\right)$

is a nonzero constant. Then $F$ is bijective, and its inverse is again a polynomial map.

Status: FALSE for every $n \ge 3$, and open for $n = 2$. Ott-Heinrich Keller posed the $n$-variable conjecture in 1939, and it is number $16$ on Smale's 1998 list of problems for the century. On 19 July 2026 Levent Alpöge announced an explicit three-variable counterexample, found with the assistance of Anthropic's Claude Fable 5. Write $F = (P, Q, R) : \mathbb{C}^3 \to \mathbb{C}^3$ with

$P = (1 + xy)^3 z + y^2 (1 + xy)(4 + 3xy), \qquad Q = y + 3x(1 + xy)^2 z + 3xy^2(4 + 3xy), \qquad R = 2x - 3x^2 y - x^3 z.$

The Jacobian determinant of $F$ is the constant $-2$, so the hypothesis holds; but

$F\left(0, 0, -\tfrac{1}{4}\right) = F\left(1, -\tfrac{3}{2}, \tfrac{13}{2}\right) = F\left(-1, \tfrac{3}{2}, \tfrac{13}{2}\right) = \left(-\tfrac{1}{4}, 0, 0\right),$

so $F$ is three-to-one over that point and is not injective. Adjoining identity coordinates turns this into a counterexample in every dimension $n \ge 3$. The coefficients are rational, so the conjecture fails over every field of characteristic zero. The case $n = 1$ is elementary and true. The two-variable case, the plane Jacobian conjecture, is still open; it is older than Keller's general form, having been stated by Ludwig Kraus in 1884.

Characteristic zero remains essential to the surviving question: in characteristic $p$ the map $x \mapsto x - x^{p}$ has Jacobian determinant $1$ and is not injective, so nothing is being conjectured there.

Neither proved nor disproved in this library. Nothing here depends on the conjecture, and this library develops neither the commutative algebra nor the algebraic geometry in which it was attacked. The refutation is a different matter: it is a finite identity in $\mathbb{Q}[x, y, z]$, and both displayed claims above were re-checked by exact rational arithmetic during the audit of this page. What this library does not contain is the theory the question belongs to, which is why the entry stays here rather than becoming a counterexample item with a proof.

What was known before, and what is now known. The converse direction is easy and is the reason the hypothesis is the natural one: if a polynomial map has a polynomial inverse then the chain rule makes the two Jacobian determinants reciprocal polynomials, and the only units of a polynomial ring over a field are the nonzero constants, so each is constant. The deepest structural result in the hard direction was the reduction of Bass, Connell and Wright (1982): the conjecture in all dimensions follows from the special case $F = X + H$ with $H$ homogeneous of degree $3$ and nilpotent Jacobian matrix, so bounding the degree never helped. The conjecture was also known to be stably equivalent to the Dixmier conjecture on endomorphisms of the Weyl algebra (Tsuchimoto 2005; Belov-Kanel and Kontsevich 2007), the Dixmier conjecture for $A_n$ implying the Jacobian conjecture in $n$ variables. That implication now runs backwards as a refutation: the three-variable counterexample shows the Dixmier conjecture is false for $A_n$ for every $n \ge 3$, while it stays open for $A_1$ and $A_2$ exactly because the plane Jacobian conjecture does.

A neighbouring statement that was already FALSE. Weakening "constant nonzero" to merely "nowhere zero" over $\mathbb{R}$ destroys the conclusion, and this was known long before 2026: Pinchuk (1994) constructed a polynomial map $\mathbb{R}^2 \to \mathbb{R}^2$ whose Jacobian determinant is everywhere positive and which is not injective. Pinchuk's example says nothing about the plane Jacobian conjecture, whose hypothesis is the stronger one that the determinant is constant.

Why it matters here. The inverse function theorem is in scope for this library, and it is exactly the local statement: a nonvanishing Jacobian determinant gives a local inverse. The Jacobian conjecture asked for the global polynomial upgrade, and the pair was for decades the cleanest illustration available of how much stronger a global statement is than its local counterpart. It is now an illustration of something else as well, which is why the entry is worth keeping rather than deleting: a question can stand for eighty-seven years, be believed by specialists in the plane case and disbelieved in high dimensions, and then be settled by a counterexample a reader can verify with nothing beyond the definition of a determinant.

Gauss-Legendre / Brent-Salamin algorithm. Set

$a_0 = 1, \qquad b_0 = \tfrac{1}{\sqrt{2}}, \qquad t_0 = \tfrac{1}{4}, \qquad p_0 = 1,$

and iterate

$a_{n+1} = \frac{a_n + b_n}{2}, \quad b_{n+1} = \sqrt{a_n b_n}, \quad t_{n+1} = t_n - p_n (a_n - a_{n+1})^2, \quad p_{n+1} = 2 p_n.$

Then

$\frac{(a_{n+1} + b_{n+1})^2}{4 t_{n+1}} \longrightarrow \pi,$

and the convergence is quadratic: the number of correct digits roughly doubles at each step, so about $25$ iterations already give tens of millions of digits.

Status: settled, but outside this library's stack. Brent and Salamin published the algorithm independently in 1976, and its correctness is a theorem. It is not open. It is also not reachable here: the proof rests on Gauss's arithmetic-geometric mean and its identification with a complete elliptic integral of the first kind, on the companion integral of the second kind, and on Legendre's relation

$E(k)K(k') + E(k')K(k) - K(k)K(k') = \frac{\pi}{2},$

none of which this library develops.

Not proved in this library. The algorithm is recorded, not derived, and no page here may present it as established.

What is known, and what would prove it here. Everything about it is known; the only obstacle is prerequisite. What would discharge this item is an elliptic integral track: the AGM iteration and its quadratic convergence, the identity $M(1,k') = \pi / (2K(k))$ expressing the AGM through the complete elliptic integral $K$, the second complete integral $E$, and Legendre's relation. That is a substantial classical development in its own right, and it belongs to a page this library does not yet have.

Why it matters here. This library defines $\pi$ analytically and will prove the elementary series and product formulas for it, including the Leibniz series and the Machin-type arctangent formulas. None of those converges as fast as the AGM iteration, and they do not even all converge at the same rate as each other. The Machin-type arctangent series converge linearly: the error falls by a fixed factor per term, so each further digit costs a bounded number of extra terms. The Leibniz series is worse than linear, and it is worth being exact about how much: its partial sums have error of order $1/n$, so each further decimal digit multiplies the number of terms by about ten and a hundred digits are already out of reach. The AGM algorithm converges quadratically, doubling the number of correct digits per step, and it is the reason the record computations of the 1980s and 1990s were feasible at all. Recording it prevents the $\pi$ pages from leaving the impression that the formulas they can prove are the ones that are actually used.

Ramanujan's series (1914).

$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!\,(1103 + 26390k)}{(k!)^4\, 396^{4k}},$

each term contributing roughly $8$ further correct decimal digits.

The Chudnovsky series (1988).

$\frac{1}{\pi} = 12 \sum_{k=0}^{\infty} \frac{(-1)^k (6k)!\,(13591409 + 545140134k)}{(3k)!\,(k!)^3\, 640320^{3k + 3/2}},$

each term contributing roughly $14$ further correct decimal digits. This is the series behind essentially every modern record computation of $\pi$.

Status: settled, but outside this library's stack. Both identities are proved theorems, not conjectures. They are not reachable here: they come from the theory of modular equations and modular forms, from singular values of the elliptic modulus, and, in the Chudnovsky case, from the class number one discriminant $-163$ that also produces the near-integer $e^{\pi \sqrt{163}}$. None of that machinery is developed in this library.

Not proved in this library. These identities are recorded with citations and are used nowhere.

What is known, and what would prove them here. Ramanujan stated seventeen series of this shape in his 1914 paper on modular equations and approximations to $\pi$, without proof; complete proofs were given much later, by J. M. and P. B. Borwein and independently by the Chudnovsky brothers, once the modular framework was in place. The general pattern is now understood as the Ramanujan-Sato family, indexed by levels and by the imaginary quadratic fields whose class numbers make the coefficients rational. What would discharge this item is a modular forms track: the modular group and its congruence subgroups, the $j$-invariant, complex multiplication and singular moduli. That is a long way outside a real analysis library.

Why it matters here. Together with the arithmetic-geometric mean algorithm, these series are the reason a reader should not conclude from the $\pi$ pages that this library has told the whole computational story. The formulas provable here converge slowly; the formulas actually used converge fast and rest on machinery from a different subject. Saying so explicitly is cheaper and more honest than silence.

For a real-valued $f$ in $L^2(\mathbb{R})$, the Hilbert transform is the singular integral taken as a Cauchy principal value,

$Hf(t) = \frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^{\infty} \frac{f(x)}{x - t} \, dx.$

Characterisation of $\pi$. The constant $\pi$ is the unique positive normalising factor $c$ for which the operator $f \mapsto \frac{1}{c}\,\mathrm{p.v.}\int f(x)/(x - t)\,dx$ satisfies $H^2 = -I$, that is, for which $H$ defines a linear complex structure on the real Hilbert space of square-integrable real-valued functions on the line. Equivalently, up to a normalising factor $H$ is the unique bounded linear operator on $L^2(\mathbb{R})$ that commutes with positive dilations and anticommutes with reflection of the line, and $\pi$ is the factor that makes it unitary.

Status: settled, but outside this library's stack. This is classical, not open. It is out of reach here because it needs the deferred measure and integration track (Lebesgue measure, the space $L^2$, principal values of singular integrals) and the deferred functional analysis track (bounded operators on a Hilbert space, unitarity, complex structures) at the same time.

Not proved in this library. The characterisation is recorded and cited, and no page here may use it.

What is known, and what would prove it here. Everything is known; the obstacle is again purely prerequisite. Discharging this item requires both deferred analysis tracks: Lebesgue measure and $L^p$ spaces on one side, and bounded operators, the Plancherel theorem and the Fourier-multiplier description of $H$ on the other. On the Fourier side $H$ is multiplication by $\pm i \,\mathrm{sgn}(\xi)$, the sign depending on which of the two standard sign conventions is taken for the kernel (the one displayed above, with $x - t$ in the denominator, gives $+i \,\mathrm{sgn}(\xi)$; writing the kernel as $t - x$ instead gives $-i \,\mathrm{sgn}(\xi)$, and the two operators differ by a sign). Either way the multiplier has modulus one and squares to $-1$, so $H$ is unitary and $H^2 = -I$ is immediate, which is why the choice of convention does not affect the characterisation. Once those tracks exist the proof is short, which is exactly why the item is a prerequisite problem and not a difficulty problem.

Why it matters here. This library treats the many equivalent characterisations of $\pi$ as a subject in its own right: the least positive zero of the sine, half the period of the complex exponential, the area of the unit disc, the value of Wallis's product, and so on. That collection is supposed to be a complete answer to "what is $\pi$", and it cannot be, because at least one of the standard characterisations lives in a subject the library does not build. Recording this one keeps the claim of completeness honest, and marks precisely which two tracks would have to exist before it could be added.

A Dowker space is a normal Hausdorff space $X$ such that $X \times [0,1]$ is not normal. By Dowker's theorem (1951) these are exactly the normal spaces that fail to be countably paracompact, so they are the witnesses that normality is not preserved by so much as multiplying with the unit interval.

Question. Does ZFC prove that a Dowker space of cardinality $\aleph_1$ exists?

Status: open. Dowker spaces themselves exist in ZFC, but only three constructions are known: M. E. Rudin's of 1971, of cardinality $\aleph_\omega^{\aleph_0}$; Balogh's of 1996, of cardinality the continuum; and the Kojman-Shelah space of 1998, of cardinality $\aleph_{\omega+1}$, obtained from pcf theory as a subspace of Rudin's. Small ones exist under extra axioms: a Dowker space of cardinality $\aleph_1$ can be built under the continuum hypothesis (Juhász, Kunen and Rudin, 1976), from the existence of a Luzin set (Todorcevic), and from the guessing principle $\clubsuit$ (de Caux, 1977; note $\diamondsuit$ implies $\clubsuit$, so it suffices too). Whether ZFC alone suffices is not known; this is Conjecture 4 of Rudin's 1990 problem list, and it is still open.

Not proved in this library, and not proved anywhere. The library now develops the required general-topology background, but it does not build a Dowker-space construction or the forcing and independence machinery needed to analyse the $\aleph_1$ question.

What is known, and what would settle it. Settling it means either a ZFC construction of a Dowker space of size $\aleph_1$, or a model of ZFC containing no such space. The consistency of the negative side is what the extra-axiom constructions do not rule out, and it is why the question is genuinely open rather than merely unresolved by the current constructions. Work since has gone on widening the hypotheses that suffice rather than removing them: Rinot, Shalev and Todorcevic derive the relevant guessing principle at $\aleph_1$ from the stick principle, from $\diamondsuit(\mathfrak{b})$, and from the existence of a Luzin set. Note the difference in status from the ambient theory: that a ZFC Dowker space exists at all was settled in 1971, and the open part is entirely about how small it can be forced to be.

Why it matters here. The library's separation-axiom material has to record that normality is badly behaved: not hereditary, not productive, and not even stable under multiplication by $[0,1]$. Dowker spaces are the canonical witness for the last of these, and they are also a clean example of a question whose answer is a cardinal rather than a yes or no. Recording it here keeps that material from asserting anything about the smallest such space.

A Suslin line is a linearly ordered set whose order topology satisfies the countable chain condition (every family of pairwise disjoint nonempty open intervals is countable) but which is not separable.

The result recorded here. If $L$ is a Suslin line then the product $L \times L$ fails the countable chain condition.

Status: settled, and outside this library's stack. This is Lemma 4.3 of Chapter II of Kunen, Set Theory: An Introduction to Independence Proofs, stated there as "if $X$ is a Suslin line, $X^2$ is not c.c.c.", with the definition of a Suslin line given in Definition 4.1 of the same section exactly as above. It is a plain ZFC theorem, and it asserts nothing about whether a Suslin line exists: only what follows if one does. It is not proved here. The library now develops order topologies and transfinite recursion through $\omega_1$, but it has not authored Kunen's specialised Suslin-line construction and square argument.

History of this entry, kept deliberately. Until the audit of 2026-07-26 this item recorded the statement as unverified: the claim had been encountered, flagged for checking against Kunen, and not checked, and the item accordingly asserted nothing. The check has now been carried out against exactly that reference, and the claim is a theorem, in the exact place the note said to look. The item id still contains the word "unverified" because ids in this library are immutable; the status above is what holds.

Not proved in this library. The result is recorded and cited, not derived, and no page here may present it as established from anything on these pages. Nothing in this library depends on it.

The proof, and why it is out of reach rather than hard. Kunen's argument is a recursion of length $\omega_1$: one picks $a_\alpha < b_\alpha < c_\alpha$ in $L$ with both intervals $(a_\alpha, b_\alpha)$ and $(b_\alpha, c_\alpha)$ nonempty and with $(a_\alpha, c_\alpha)$ avoiding every $b_\xi$ already chosen, which is possible because $L$ is not separable. The $\omega_1$ many open rectangles $V_\alpha = (a_\alpha, b_\alpha) \times (b_\alpha, c_\alpha)$ are then nonempty and pairwise disjoint, so $L^2$ is not ccc. The obstacle here is the setting, not the difficulty: separability is developed later in Separability: the existence of an at most countable dense subset ↗, but this item still records rather than proves Kunen's specialised $\omega_1$-length recursion.

Relation to the partial-order form. The Suslin hypothesis is independent of ZFC, in both directions ‡ records, with citations, that a Suslin line yields a ccc partial order whose square is not ccc, which is the Suslin tree viewed as a forcing. The statement here is the topological one about the square of the line itself, and the two agree: the nonempty open subsets of a ccc space, ordered by inclusion, form a ccc partial order, so the rectangles above give the partial-order statement too.

What is still not asserted. The existence of a Suslin line is independent of ZFC: the consistency of Suslin's Hypothesis, that no Suslin line exists, is due to Solovay and Tennenbaum (1971) by iterated ccc forcing, while $\diamondsuit$ implies one exists, so one exists in the constructible universe. Nothing here asserts that a Suslin line exists, and therefore nothing here asserts outright that the countable chain condition fails to be productive. That conclusion is conditional on there being a Suslin line, which is precisely why "the product of two ccc spaces is ccc" is not decided by ZFC.

Why it matters here. Ccc arguments will appear in the library's topology material, and productivity of the countable chain condition is exactly the point at which a plausible-sounding claim silently imports an independence result. Any page wanting an unconditional ZFC counterexample about the countable chain condition should use the Cantor cube $\{0,1\}^{\kappa}$ for $\kappa$ larger than the continuum, which is ccc and not separable and needs no independence result at all, and leave the Suslin line to this item.

Work in ZF, without the axiom of choice. Write HB for the Hahn-Banach theorem: if $p$ is a sublinear functional on a real vector space $X$ and $f$ is a linear functional on a subspace of $X$ dominated by $p$, then $f$ extends to a linear functional on all of $X$ still dominated by $p$.

Question. Does HB imply that $\mathbb{R}$, as a vector space over $\mathbb{Q}$, has a basis?

Status: open. It is recorded as an open question in Howard and Rubin's catalogue of consequences of the axiom of choice, as the implication from their form for Hahn-Banach to their form for a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$. No proof of the implication and no model of ZF separating the two is known.

Not proved in this library, and not proved anywhere. This library does not develop functional analysis and does not build models of ZF, so neither half of the question is reachable here; both belong to deferred tracks. Nothing on any page depends on the answer.

What is known, and what would settle it. The endpoints are well understood. Full choice gives a Hamel basis, since "every vector space has a basis" is equivalent to the axiom of choice (The Axiom of Choice ↗), the standard proof running through Zorn's lemma (Zorn's lemma ↗). In the other direction, granted the consistency of ZF, HB is not a theorem of ZF + DC. The cheapest route to that, and the one this item relies on, does not go through Lebesgue measure. HB applied to a nonzero element of $\ell^\infty / c_0$ produces a nonzero linear functional on that space, and from such a functional one gets a set of reals without the Baire property. Shelah (1984) showed that Solovay's inaccessible can be dispensed with for the Baire property, so the consistency of ZF alone yields a model of ZF + DC in which every set of reals has the Baire property; in that model $(\ell^\infty / c_0)^*$ is trivial, so HB fails there. This is worth spelling out because the obvious argument is more expensive: HB also implies, in ZF, the existence of a non-Lebesgue-measurable set (Foreman and Wehrung, 1991) and indeed the Banach-Tarski paradox (Pawlikowski, 1991), but a model of ZF + DC in which every set of reals is measurable costs an inaccessible cardinal, so that route would only give the unprovability of HB relative to a large cardinal.

HB is also, granted Con(ZF), strictly weaker than choice: the Boolean prime ideal theorem implies it outright (Luxemburg, 1969), and BPI does not imply the axiom of choice (Halpern and Lévy, 1971), so neither does HB. Pincus (1974) proved the sharper separation that HB does not imply BPI, refuting the prevailing conjecture of the 1960s. All of these are relative-consistency results and nothing stronger. A Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$ likewise yields a non-measurable set. So the two statements sit strictly between ZF and AC, on those cited results and under the consistency of ZF, and the question is how they are ordered with respect to each other. Settling it means either deriving a Hamel basis from HB in ZF, or producing a model of ZF in which HB holds and $\mathbb{R}$ has no basis over $\mathbb{Q}$.

The nearest recent progress is a separation of the two classical consequences of a Hamel basis from each other: Larson and Shelah (2026) construct a model of ZF + DC containing a discontinuous additive endomorphism of $\mathbb{R}$ but no Hamel basis for $\mathbb{R}$. That does not touch HB, but it shows the two targets in this and the companion question are genuinely different targets and not notational variants.

A note on the reference. The form numbers used for these questions in the working notes are $52$ for Hahn-Banach, $367$ for a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$ and $366$ for a discontinuous additive function, taken from the Howard-Rubin numbering. These three numbers are unverified. The Consequences of the Axiom of Choice project database that hosted the searchable numbering, and the later mirror of it, both fail to answer as of 2026-07-26, and no other online source consulted lists the numbering, so they could not be re-checked; the book remains the reference and the numbers should be treated as a pointer into it rather than as a verified citation.

Why it matters here. This library keeps an explicit ledger of what each result costs in choice, and the ledger is supposed to be exact. This entry is a place where exactness is impossible: the cost of one of the most-used theorems in analysis, measured against one of the most-used pathologies in analysis, is not known. That is worth stating rather than rounding off to "both need choice".

Work in ZF, without the axiom of choice, and write HB for the Hahn-Banach theorem.

Question. Does HB imply the existence of a discontinuous additive function $f : \mathbb{R} \to \mathbb{R}$, that is, a solution of Cauchy's functional equation $f(x + y) = f(x) + f(y)$ that is not of the form $f(x) = cx$?

Status: open. Like the companion question about a Hamel basis, this is recorded as open in Howard and Rubin's catalogue of consequences of the axiom of choice, as the implication from their form for Hahn-Banach to their form for a discontinuous additive function on the line.

Not proved in this library, and not proved anywhere. No page here depends on the existence of a pathological solution of Cauchy's equation, and this library develops neither functional analysis nor ZF model construction.

What is known, and what would settle it. For an additive $f : \mathbb{R} \to \mathbb{R}$, being continuous, being linear over $\mathbb{R}$, being measurable, being bounded on some set of positive measure and being bounded on some interval are all the same condition, so a discontinuous additive function is an extremely wild object: its graph is dense in the plane. The axiom of choice (The Axiom of Choice ↗) produces one immediately from a Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$, by choosing a $\mathbb{Q}$-linear map that is not $\mathbb{R}$-linear. In the other direction, granted the consistency of ZF, ZF + DC cannot produce one: in Solovay's model, and in Shelah's 1984 strengthening that removes the inaccessible cardinal, every set of reals has the Baire property, and then every additive $f : \mathbb{R} \to \mathbb{R}$ is continuous. Shelah's version is what makes the consistency hypothesis just Con(ZF): Solovay's model on its own would need an inaccessible. So the statement sits strictly between ZF and AC, exactly as HB does, and the question is again how the two are ordered. Settling it means a ZF derivation from HB, or a model of ZF with HB and no discontinuous additive function.

Two nearby results sharpen what is at stake. Larson and Shelah (2026) build a model of ZF + DC with a discontinuous additive endomorphism of $\mathbb{R}$ but no Hamel basis for $\mathbb{R}$, so this consequence is strictly weaker than the Hamel basis in that setting and the two open questions are genuinely distinct. And the $\mathbb{R}$-linear analogue is at least as expensive: in the same ZF + DC model in which every set of reals has the Baire property, every linear functional on a Banach space is continuous, so the existence of a discontinuous linear functional on an infinite-dimensional Banach space is itself not provable in ZF + DC. It follows from the axiom of choice, but it is not known to this library's sources to be equivalent to it, and nothing here claims that it is. The function asked about above is only $\mathbb{Q}$-linear, which is what leaves room for it to be cheaper than either.

Why it matters here. A discontinuous additive function is the smallest and most-cited pathology in real analysis whose existence is not a theorem of ZF. Any statement of the form "the only additive functions are the linear ones" is a statement about the ambient set theory, not about the reals, and the library's choice ledger is where that has to be recorded. This item records that the exact price of the pathology, measured against Hahn-Banach, is unknown.