Kazimierz Kuratowski

How to use this course

Kuratowski’s name is attached to more standard results than almost any other twentieth-century topologist. Students meet the Kuratowski closure axioms in a first topology course, the Kuratowski ordered pair in a first set theory course, the Kuratowski–Zorn lemma in algebra, and Kuratowski’s theorem in graph theory. Each is short to state, and each changed how its subject was written. This course presents those results together with the life of the man who, for half a century, was one of the organizers of the Polish school of mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order is roughly chronological: closure (thesis of 1921, published 1922), pairs and finiteness (1920–1921), the maximal principle (1922), fixed points (1929), planarity (1930), the logic of projective sets (1931) and later work. Companion modules on Wacław Sierpiński, Stefan Banach, Alfred Tarski and Stanisław Ulam cover the same school from other sides.

Timeline

Warsaw and Glasgow, 1896–1915

Kazimierz Kuratow was born in Warsaw on 2 February 1896. The city was then the capital of Congress Poland, ruled by the Russian Empire. His family was assimilated Jewish; his father, Marek Kuratow, was a well-known Warsaw barrister, and his mother was Róża Karzewska. He took the name Kuratowski in 1931, and this course uses that name throughout.

He finished secondary school at the General Paweł Chrzanowski school in Warsaw. Higher education in Warsaw meant study in Russian at the Imperial University, which many Polish families boycotted. So in October 1913 he went abroad and matriculated at the University of Glasgow to study engineering. He did well: at the end of his first year he won the class prize in mathematics.

When war broke out in August 1914 he could not return to Scotland. The war that cut off his engineering career also opened a new path. In the summer of 1915 the Russian army withdrew from Warsaw, and the German occupation authorities allowed the university to reopen as a Polish institution. Kuratowski decided to study mathematics there. Glasgow did not forget him; decades later it gave him an honorary doctorate.

The new Polish university, 1915–1921

The University of Warsaw reopened in November 1915, and Kuratowski was among its first mathematics students. Two young topologists shaped him. Zygmunt Janiszewski had written a French thesis on continua and was arguing that Polish mathematics should concentrate on a few new fields, set theory, topology and logic, and found its own journal. Stefan Mazurkiewicz ran the seminar in topology. Sierpiński joined them in 1918 after returning from wartime internment in Russia.

Kuratowski’s first paper, on definitions in mathematics, appeared in 1917. He graduated in 1919. When Fundamenta Mathematicae began in 1920 he was already one of its authors: its first volume carried his paper on the notion of a finite set (Lesson 05) and a joint paper with Janiszewski on continua. Janiszewski died in January 1920, before the volume appeared.

The thesis

He received his doctorate in the autumn of 1921 with Mazurkiewicz as supervisor; the Mathematics Genealogy Project lists Janiszewski as a second advisor. The thesis had two parts. The first built topology from four axioms on a closure operation. The second studied continua irreducible between two points, the subject of Janiszewski’s own thesis, and settled questions about them raised by Charles de la Vallée Poussin. The first part, slightly revised, appeared in Fundamenta in 1922 as Sur l’opération \(\overline{A}\) de l’Analysis Situs. It is the source of the closure axioms in Lesson 04 and has been cited in hundreds of later papers.

Warsaw and Lwów, 1921–1934

In 1923 Kuratowski became deputy professor of mathematics at the University of Warsaw. These were productive years. He wrote on set theory, on the foundations of topology, and with Bronisław Knaster on the strange connected sets that filled the early volumes of Fundamenta. His 1922 paper on eliminating transfinite numbers from proofs gave the maximal principle of Lesson 06.

In 1927 he became full professor at the Lwów Polytechnic. He headed its mathematics department until 1933 and served twice as dean. Lwów was then the home of Banach and Hugo Steinhaus and the centre of functional analysis. Kuratowski joined the circle that met in the Scottish Café and worked with Banach on measure problems. In 1929 they proved that, under the continuum hypothesis, there is no countably additive measure defined on all subsets of \([0,1]\) that vanishes on points and gives the whole interval measure 1. Ulam later removed the hypothesis.

In his first term, in 1927, Kuratowski taught a set theory course to about fifteen students. One of them was the eighteen-year-old Stanisław Ulam, who solved an open problem from the course. Ulam recalled that Kuratowski walked home with him after lectures, talking mathematics, and pushed him to write up his results. Ulam took his doctorate under Kuratowski at the Polytechnic in 1933, Kuratowski’s first doctoral student. Kuratowski married in 1929, and his daughter Zofia was born in 1931; she became a haematologist and, after 1989, a senator.

The Lwów years saw the KKM lemma (1929), the planarity theorem (1930), the paper with Tarski on logic and projective sets (1931), the Kuratowski–Ulam theorem (1932) and the first volume of Topologie (1933). In 1934 he returned to Warsaw as professor, a year before the Scottish Book was started, so he contributed no problems to it.

Closure axioms and the fourteen sets

For the general reader. What does it mean for a point to be “near” a set? Felix Hausdorff had described topology through neighbourhoods. Kuratowski showed that one operation, which adds to a set all the points touching it, carries all the information, and that four simple rules about it are enough. He then asked a puzzle-like question about this operation and found the answer 14.

The axioms

Axiom 4 implies monotonicity: if \(A\subseteq B\) then \(\overline B=\overline{A\cup B}=\overline A\cup\overline B\supseteq\overline A\). Call \(F\) closed if \(\overline F=F\). Then \(\varnothing\) and \(X\) are closed, finite unions of closed sets are closed by axiom 4, and arbitrary intersections are closed: if \(F=\bigcap F_i\) then \(\overline F\subseteq\overline{F_i}=F_i\) for every \(i\). So the complements of closed sets form a topology. Conversely, in any topological space the operation “smallest closed superset” satisfies the four axioms.

Kuratowski’s formulation was algebraic: the axioms are identities in the Boolean algebra of subsets of \(X\) with one extra operation. This point of view led directly to the closure algebras of John McKinsey and Tarski in the 1940s and to the link between topology and modal logic.

Closure and complement

Write \(k\) for closure and \(c\) for complement, and \(i=ckc\) for interior. Apply \(k\) and \(c\) to a set \(A\) in every order. How many different sets can appear?

Proof sketch. Because \(c^2=1\) and \(k^2=k\), every word reduces to an alternating word in \(k\) and \(c\). The key identity is \(kiki=ki\), equivalently \(kckckck=kck\). Granting it, every alternating word of length at least 7 shortens, and the words that remain, applied to \(A\), give at most the seven sets

together with their seven complements. To prove \(kiki=ki\), let \(B=kiA\). Then \(iB\subseteq B\), so \(kiB\subseteq kB=B\). In the other direction, \(iA\) is open and \(iA\subseteq kiA=B\), so \(iA\subseteq iB\) and therefore \(B=kiA\subseteq kiB\).

Ordered pairs and finite sets

For the general reader. If everything in mathematics is a set, then an ordered pair, a function and the notion of “finite” must all be sets or properties of sets. Kuratowski gave the definitions of the pair and of finiteness that made this program clean. His ordered pair is the one every set theory textbook now uses.

The ordered pair (1921)

Norbert Wiener in 1914 had defined an ordered pair as \(\{\{\{a\},\varnothing\},\{\{b\}\}\}\), and Hausdorff gave another definition the same year that used two auxiliary objects distinct from \(a\) and \(b\). In Sur la notion de l’ordre dans la théorie des ensembles (Fundamenta 2, 1921) Kuratowski proposed the simpler

Proof. One direction is trivial. Suppose \(\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}\). If \(a=b\), the left side is \(\{\{a\}\}\), so \(\{c\}=\{c,d\}=\{a\}\), giving \(c=d=a=b\). If \(a\ne b\), the left side has one singleton and one two-element set, so the same is true of the right side, \(c\ne d\), and matching sizes gives \(\{a\}=\{c\}\) and \(\{a,b\}=\{c,d\}\). Then \(a=c\), and \(b\in\{c,d\}\) with \(b\ne a=c\) forces \(b=d\).

With pairs in hand, a relation is a set of ordered pairs and a function is a relation in which each first coordinate occurs once. The cartesian product \(A\times B\) is a subset of \(\mathcal P(\mathcal P(A\cup B))\), so its existence follows from the power set and separation axioms. MacTutor notes that this made separate notions of function, such as those of Gottlob Frege, Charles Peirce and Ernst Schröder, unnecessary.

Finite sets (1920)

Defining “finite” as “in bijection with \(\{1,\dots,n\}\)” presupposes the natural numbers. Kuratowski’s definition uses only sets.

Equivalently, \(A\) is in the smallest family of subsets of \(A\) containing \(\varnothing\) and all singletons and closed under binary unions. Richard Dedekind had called a set finite if it is not in bijection with a proper subset. In ZF every Kuratowski-finite set is Dedekind-finite; the converse requires a weak form of the axiom of choice. Kuratowski finiteness is the version used in topos theory and constructive mathematics, where choice is unavailable.

The maximal principle

For the general reader. Many existence proofs in algebra and analysis used transfinite induction: well-order everything, then build the object step by step through infinite ordinals. In 1922 Kuratowski showed that a single statement about maximal elements can replace these constructions. Max Zorn rediscovered it in 1935, and it became known as Zorn’s lemma; in Poland and Russia it is called the Kuratowski–Zorn lemma.

The paper was Une méthode d’élimination des nombres transfinis des raisonnements mathématiques (Fundamenta 3, 1922). Hausdorff had earlier used a maximal principle for chains. Kuratowski’s aim was methodological: to show that ordinals could be removed from the statements and proofs of analysis and topology.

Zorn’s version (Bull. Amer. Math. Soc., 1935) used arbitrary chains and proposed the principle as an axiom for algebra. John Tukey gave it the name “Zorn’s lemma” in 1940. Over ZF the lemma is equivalent to the axiom of choice and to the well-ordering theorem.

Proof sketch from choice

Suppose \(P\) has no maximal element. Then every chain \(C\) has a strict upper bound, and the axiom of choice gives a function \(g\) choosing one, \(g(C)\notin C\). Call a well-ordered chain \(W\) a \(g\)-tower if each \(x\in W\) equals \(g(\{w\in W:w<x\})\). Any two \(g\)-towers are comparable, one being an initial segment of the other, by a short induction on the well order. Their union \(U\) is then itself a \(g\)-tower. But \(U\cup\{g(U)\}\) is a longer \(g\)-tower, so \(g(U)\in U\), a contradiction. Kuratowski’s point was that this argument mentions no ordinals.

Planar graphs

For the general reader. Three houses must each be joined to gas, water and electricity without pipes crossing. It cannot be done in the plane. Neither can five towns be joined pairwise by non-crossing roads. Kuratowski proved in 1930 that these two obstacles are the only ones: a network can be drawn without crossings unless it contains one of them in disguise.

A subdivision of a graph \(H\) is obtained by replacing edges of \(H\) with paths whose interior vertices are new. \(K_5\) is the complete graph on five vertices; \(K_{3,3}\) is the complete bipartite graph with parts of size three.

The paper is Sur le problème des courbes gauches en topologie, Fundamenta 15 (1930), 271–283. Kuratowski worked topologically, with curves in the plane. Orrin Frink and Paul Smith announced an independent proof in 1930 that they never published. Lev Pontryagin is reported to have found a proof around 1927, also unpublished, and in Russian literature the result is sometimes called the Pontryagin–Kuratowski theorem.

Necessity

For a connected plane graph with \(V\) vertices, \(E\) edges and \(F\) faces, Euler’s formula gives \(V-E+F=2\). If every face is bounded by at least \(g\) edges (\(g\ge3\)), counting edge–face incidences gives \(2E\ge gF\), so

For \(K_5\), \(g=3\) and \(10>3\cdot3=9\). For \(K_{3,3}\), which has no triangles, \(g=4\) and \(9>2\cdot4=8\). Neither is planar, and a subdivision is planar exactly when the original graph is.

Sufficiency: the shape of a proof

Take a counterexample \(G\) with the fewest edges: non-planar, with no Kuratowski subgraph. One shows first that \(G\) is 3-connected, since along a separation of size at most 2 the pieces can be embedded separately and glued. A 3-connected graph with at least five vertices has an edge \(xy\) whose contraction \(G/xy\) is still 3-connected (a lemma of Carsten Thomassen). By minimality \(G/xy\) is planar; a Kuratowski subgraph in it would lift to one in \(G\). In the embedding of \(G/xy\), delete the merged vertex; its neighbours lie on a cycle \(C\) bounding a face. Now look at how the neighbours of \(x\) and of \(y\) sit on \(C\). If they occupy separate arcs, \(x\) and \(y\) can be put back into the face and \(G\) is planar. If they interleave, three common neighbours give a subdivided \(K_5\), and an alternating pattern \(x,y,x,y\) gives a subdivided \(K_{3,3}\).

Klaus Wagner’s form led, through the work of Neil Robertson and Paul Seymour, to the graph minor theorem: every minor-closed family of graphs is characterized by finitely many forbidden minors. Algorithmically, planarity can be tested in linear time (John Hopcroft and Robert Tarjan, 1974), and a Kuratowski subgraph of a non-planar graph can also be found in linear time, which lets a program certify its answer.

The KKM lemma and fixed points

For the general reader. Stir a cup of coffee: L. E. J. Brouwer’s theorem says some point of the liquid ends where it started. In 1929 Knaster, Kuratowski and Mazurkiewicz found a proof that replaces algebraic topology with a covering statement about closed sets and a counting lemma. Their lemma later became a basic tool in economics and game theory.

Let \(\Delta_n=\{x\in\mathbb R^{n+1}:x_i\ge0,\ \sum x_i=1\}\) be the standard simplex with vertices \(e_0,\dots,e_n\). For \(I\subseteq\{0,\dots,n\}\) let \(\Delta_I\) be the face spanned by \(\{e_i:i\in I\}\).

The paper is Ein Beweis des Fixpunktsatzes für \(n\)-dimensionale Simplexe, Fundamenta 14 (1929). It used the combinatorial lemma Emanuel Sperner had published in 1928.

Proof from Sperner’s lemma

Triangulate \(\Delta_n\) with mesh less than \(\varepsilon\). Give each vertex \(v\) of the triangulation a label \(\lambda(v)=i\) with \(v\in C_i\) and \(v_i>0\). This is possible: if \(I=\{i:v_i>0\}\), then \(v\in\Delta_I\subseteq\bigcup_{i\in I}C_i\). The labelling obeys Sperner’s rule (a vertex on face \(\Delta_I\) gets a label in \(I\)), so some small simplex carries all labels \(0,\dots,n\). Its vertices are within \(\varepsilon\) of each other, and the \(i\)-labelled one lies in \(C_i\). Let \(\varepsilon\to0\); by compactness a subsequence of these simplices shrinks to a point \(x\), and since each \(C_i\) is closed, \(x\in C_i\) for all \(i\).

Brouwer from KKM

Let \(f:\Delta_n\to\Delta_n\) be continuous and set \(C_i=\{x: f(x)_i\le x_i\}\), closed by continuity. If \(x\in\Delta_I\), then \(\sum_{i\in I}x_i=1\ge\sum_{i\in I}f(x)_i\), so \(f(x)_i\le x_i\) for some \(i\in I\): the covering condition holds. A point \(x\) in every \(C_i\) has \(f(x)_i\le x_i\) for all \(i\), and both vectors sum to 1, so \(f(x)=x\).

Ky Fan extended the lemma in 1961 to infinite-dimensional “KKM maps”, and in that form it yields minimax theorems, the existence of Nash equilibria and of general economic equilibria.

Logic, category and selection

For the general reader. Descriptive set theory classifies sets of real numbers by how complicated their definitions are. Kuratowski found three tools that are now standard there: a dictionary between logical quantifiers and set operations, a version of Fubini’s theorem for “small” sets in the sense of Baire, and a theorem that lets one choose points measurably from a family of closed sets.

Quantifiers and set operations (1931)

In Les opérations logiques et les ensembles projectifs (Fundamenta 17, 1931), Kuratowski and Tarski observed that the shape of a definition predicts where a set sits in the Borel and projective hierarchies. Conjunction and disjunction correspond to intersection and union, and negation to complement. A quantifier over a countable set corresponds to a countable intersection or union, and a quantifier over the reals to a projection or the complement of a projection. Writing a definition in prenex form and counting alternations gives an upper bound on its complexity. The method is now called the Tarski–Kuratowski algorithm.

Category in products (1932)

A set is meagre if it is a countable union of nowhere dense sets, and comeagre if its complement is meagre. For \(A\subseteq X\times Y\) and \(x\in X\), let \(A_x=\{y:(x,y)\in A\}\).

This is the topological analogue of Fubini’s theorem for null sets, and like Fubini it fails without a regularity hypothesis on \(A\).

Measurable selection (1965)

The theorem is used throughout stochastic control, mathematical economics and optimization, wherever one must choose an optimal action measurably.

Metric spaces in Topologie

For the general reader. Kuratowski’s Topologie (volume I, 1933; volume II, 1950) was for decades the standard reference on metric spaces and the topology of the plane. This lesson takes three of its tools: a way to place any metric space inside a space of functions, a number that measures how far a set is from being compact, and a notion of limit for sequences of sets.

The embedding

By the triangle inequality \(|\phi_x(z)|\le d(x,x_0)\), so \(\phi_x\) is bounded. Also \(|\phi_x(z)-\phi_y(z)|=|d(x,z)-d(y,z)|\le d(x,y)\), with equality at \(z=y\). Hence \(\|\phi_x-\phi_y\|_\infty=d(x,y)\). Since \(C_b(X)\) is complete, the closure of the image is a completion of \(X\).

Measuring non-compactness (1930)

For a bounded set \(A\) in a metric space, Kuratowski defined

So \(\alpha(A)=0\) exactly when \(A\) is totally bounded. In a complete space that means \(\overline A\) is compact.

Proof sketch. Pick \(x_n\in F_n\). For each \(k\), all but finitely many \(x_n\) lie in \(F_k\), which is covered by finitely many sets of diameter about \(\alpha(F_k)\). A diagonal argument extracts a Cauchy subsequence, which converges in \(X\); its limit lies in every closed \(F_k\). Finally \(\alpha(F_\infty)\le\alpha(F_n)\to0\), so \(F_\infty\) is closed and totally bounded, hence compact. With \(\operatorname{diam}\) in place of \(\alpha\) this is Cantor’s intersection theorem. Gabriele Darbo used \(\alpha\) in 1955 to extend Schauder’s fixed point theorem to condensing maps.

Limits of sets

For sets \(A_n\) in a metric space, the lower limit \(\operatorname{Li}A_n\) is the set of points \(x\) such that every neighbourhood of \(x\) meets all but finitely many \(A_n\), and the upper limit \(\operatorname{Ls}A_n\) is the set of \(x\) whose every neighbourhood meets infinitely many \(A_n\). The sequence converges in Kuratowski’s sense when the two agree. This notion, also associated with Paul Painlevé, is basic in variational analysis.

Occupation and the secret university, 1934–1945

Back in Warsaw from 1934, Kuratowski headed the mathematics department of the university from 1935 and from 1936 to 1939 was secretary of the mathematics committee of the Council of Science and Applied Sciences. His doctoral students included Samuel Eilenberg (1936), later a founder of category theory with Saunders Mac Lane, and Andrzej Mostowski (1938), who also worked closely with Tarski and became the leading Polish logician of the post-war period.

Germany invaded Poland in September 1939. The occupiers closed Polish universities and secondary schools and banned higher education for Poles. A clandestine University of Warsaw continued in private flats, and Kuratowski taught in it throughout the occupation. Ulam, who heard his account after the war, recorded that he had several narrow escapes from arrest by the Gestapo. Many colleagues did not survive. In Lwów, Włodzimierz Stożek, Antoni Łomnicki and Kazimierz Bartel were among the professors shot by the Germans in July 1941. Mazurkiewicz died in 1945, shortly after the war. Ulam later wrote that Kuratowski told him nothing could upset him after what he had lived through.

Rebuilding, last years and legacy, 1945–1980

In February 1945 Kuratowski was lecturing again at the reopened University of Warsaw. Ulam judged that the reconstruction of Polish mathematics became the central work of his later life. He was president of the Polish Mathematical Society from 1946 to 1953. From 1948 he directed the new State Institute of Mathematics, which became part of the Polish Academy of Sciences in 1952, and he remained its director until 1967 and head of its topology section until his death. He was vice-president of the Academy from 1957 to 1968 and vice-president of the International Mathematical Union from 1963 to 1966. From 1952 he was editor-in-chief of Fundamenta Mathematicae.

He wrote steadily: more than 170 papers and several books. Topologie II appeared in 1950. With Mostowski he wrote Teoria mnogości (1952), later translated as Set Theory. His Introduction to Set Theory and Topology (1952) was translated into English, French, Spanish and Bulgarian. His memoir of Polish mathematics, published in Polish in 1973 and in English in 1980 as A Half Century of Polish Mathematics, remains a primary source on the interwar school. His later doctoral students included Jerzy Jaroń (1958), Stanisław Mrówka (1959), Ryszard Engelking (1961), Monika Karłowicz (1966) and Józef Krasinkiewicz (1971). Engelking’s General Topology continued the line of Topologie.

Kuratowski died in Warsaw on 18 June 1980, aged 84. Notes for an autobiography, which he had begun publishing in 1979, appeared as a book in 1981, prepared by his daughter Zofia Kuratowska.

Honours

He was elected to the Warsaw Scientific Society (1929), the Polish Academy of Learning (1945) and the Polish Academy of Sciences (1952), and abroad to the academies of the USSR, Hungary, Austria, the German Democratic Republic, Argentina and Italy (the Accademia dei Lincei); he became a Fellow of the Royal Society of Edinburgh in 1966. He received honorary doctorates from Glasgow, Paris, Prague and Wrocław, and the gold medal of the Czechoslovak Academy of Sciences. In 1981 the Institute of Mathematics, the Polish Mathematical Society and his daughter founded the Kuratowski Prize for mathematicians under 30.

Legacy

• Topology: the closure axioms and Topologie fixed the language of general topology, and the measure of non-compactness and set convergence remain working tools in analysis.
• Set theory: the ordered pair, finiteness without numbers, and the maximal principle that replaces transfinite induction.
• Graph theory: the first forbidden-subgraph characterization, the ancestor of the graph minor theorem.
• Descriptive set theory: the Tarski–Kuratowski algorithm, the Kuratowski–Ulam theorem and measurable selection.
• Institutions: two generations of Polish mathematicians, from Ulam, Eilenberg and Mostowski to Engelking, and the Institute of Mathematics he led for nearly twenty years.

Connections

People linked to Kazimierz Kuratowski across this collection. A name without a link has no page here yet. See Kazimierz Kuratowski in the network of all 337 lives →

Teachers and advisors

• Stefan Mazurkiewicz 1921
• Zygmunt Janiszewski 1915-1920

Students

• Stanislaw Ulam 1933
• Andrzej Mostowski 1938
• Jerzy Jaroń 1958
• Józef Krasinkiewicz 1971
• Monika Karłowicz 1966
• Ryszard Engelking 1961
• Samuel Eilenberg 1936
• Stanisław Mrówka 1959

Influenced by

• Felix Hausdorff

Influenced

• Klaus Wagner
• Max Zorn

Colleagues and collaborators

• Alfred Tarski 1931
• Stefan Banach 1927-1934
• Wacław Sierpiński 1918-1969
• Bronisław Knaster 1921-1929
• Czesław Ryll-Nardzewski 1965

Also mentioned in this life

• L. E. J. Brouwer
• Georg Cantor
• Augustin-Louis Cauchy
• Richard Dedekind
• Leonhard Euler
• Gottlob Frege
• David Hilbert
• Charles Sanders Peirce
• Norbert Wiener

Fields

• topology 9 lives
• set theory 12 lives
• graph theory 4 lives

On CoursesHub

• Topology in Physics · Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Kazimierz Kuratowski, MacTutor History of Mathematics.
2. S. Ulam, Kazimierz Kuratowski (1896–1980), obituary, reproduced at MacTutor.
3. Kazimierz Kuratowski, Wikipedia.
4. Kazimierz Kuratowski, Mathematics Genealogy Project.
5. K. Kuratowski, A Half Century of Polish Mathematics: Remembrances and Reflections, Pergamon, 1980.
6. S. Ulam, Adventures of a Mathematician, Scribner, 1976.

Mathematics

1. K. Kuratowski, Sur la notion d’ensemble fini, Fund. Math. 1 (1920).
2. K. Kuratowski, Sur la notion de l’ordre dans la théorie des ensembles, Fund. Math. 2 (1921), 161–171.
3. K. Kuratowski, Sur l’opération \(\overline{A}\) de l’Analysis Situs, Fund. Math. 3 (1922), 182–199.
4. K. Kuratowski, Une méthode d’élimination des nombres transfinis des raisonnements mathématiques, Fund. Math. 3 (1922).
5. B. Knaster, C. Kuratowski, S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929), 132–137.
6. K. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283.
7. K. Kuratowski, A. Tarski, Les opérations logiques et les ensembles projectifs, Fund. Math. 17 (1931).
8. K. Kuratowski, S. Ulam, Quelques propriétés topologiques du produit combinatoire, Fund. Math. 19 (1932), 247–251.
9. K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. 13 (1965), 397–403.
10. K. Kuratowski, Topology, vols. I–II, Academic Press and PWN, 1966–1968 (translation of Topologie).
11. K. Kuratowski, A. Mostowski, Set Theory, North-Holland and PWN.
12. M. Zorn, A remark on method in transfinite algebra, Bull. Amer. Math. Soc. 41 (1935).
13. C. Thomassen, Kuratowski’s theorem, J. Graph Theory 5 (1981).
14. A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.