Pavel Alexandrov

How to use this course

Pavel Alexandrov was one of the architects of topology as it is taught today. The definitions of compact and locally compact space, the one-point compactification, the nerve of a covering and the homological approach to dimension all come from his work, much of it done in the 1920s in conversation with Pavel Urysohn, Emmy Noether, L. E. J. Brouwer and Heinz Hopf. He was also a teacher of rare influence. Andrey Tikhonov, Lev Pontryagin, Aleksandr Kurosh and many others came out of his seminars, and he led the Moscow Mathematical Society for 32 years.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 07 are point-set topology and follow one another; Lessons 08 to 10 can be read independently.

Timeline

Smolensk and Moscow, 1896–1917

Pavel Sergeyevich Alexandrov was born on 7 May 1896 in Bogorodsk, a town east of Moscow now called Noginsk. His father, Sergei Aleksandrovich Alexandrov, was a doctor who had trained at Moscow University; about a year after Pavel’s birth he took a post at the state hospital in Smolensk, and the family settled there. His mother, Tsezariya Akimovna, taught him French and German at home. The house was full of music, and music remained one of his lifelong passions.

At the Smolensk gymnasium his mathematics teacher, Aleksandr Eiges, drew him toward the foundations of geometry and non-Euclidean geometry. He finished school in 1913 as the top pupil, with a gold medal, and entered Moscow University that autumn.

Luzin’s first student

At Moscow, Vyacheslav Stepanov encouraged him to join Dmitri Egorov’s seminar, and in his second year he began working with Nikolai Luzin, who had just returned from Paris full of the new descriptive set theory of Émile Borel, René Baire and Henri Lebesgue. Luzin’s school, known to its members as Lusitania, became the centre of Moscow mathematics. Alexandrov was its first research student.

In 1915 he proved his first major result: every uncountable Borel set of real numbers contains a perfect subset, and therefore has the cardinality of the continuum. Felix Hausdorff found the same result independently. To prove it, Alexandrov introduced a set-theoretic operation that his fellow student Mikhail Suslin then used to define the analytic sets (Lesson 04). Luzin next set him the continuum hypothesis itself. He could not make progress, which is no surprise: Paul Cohen showed in 1963 that the hypothesis cannot be decided from the usual axioms. The failure hit him hard. He graduated in 1917 and, for a time, left mathematics.

The theatre and the return, 1917–1922

The years of revolution and civil war took Alexandrov away from Moscow. He worked as a theatre producer in Novgorod-Severskii and then in Chernigov, in Ukraine, where he also taught languages. In 1919 he was briefly jailed by one of the forces fighting over the region. His interest in literature and the stage never left him.

Conversations with Luzin’s circle and a visit to Moscow persuaded him to return to mathematics. In 1920–21 he taught at the new university in Smolensk while preparing for the Moscow examinations, and in 1921 he became a lecturer at Moscow University. He had moved away from Luzin’s descriptive set theory and toward a subject that barely existed yet: the general theory of topological spaces. Hausdorff’s Grundzüge der Mengenlehre (1914) had defined spaces by neighbourhood axioms, but almost nothing was known about which properties of the real line or of Euclidean space survive in that generality.

Urysohn, Göttingen and Hopf, 1922–1929

In the summer of 1922 Alexandrov and Pavel Urysohn, two years younger and also from the Luzin circle, began working together at a dacha in Bolshevo near Moscow. In a few months they created the theory of compact spaces. Their notion, which Alexandrov called bicompactness, is the open-cover definition of compactness used today (Lesson 05). They also defined local compactness and found the first general metrization theorem (Lesson 07).

In 1923 and 1924 the two travelled to Göttingen, where David Hilbert, Richard Courant and Emmy Noether received their results with enthusiasm. Noether became a close friend of Alexandrov from that first summer. In 1924 they visited Hausdorff in Bonn and Brouwer in Blaricum, then went to France. On 17 August 1924 Urysohn drowned while swimming in the Atlantic at Batz-sur-Mer in Brittany. He was 26. Alexandrov, who was with him, spent much of 1925 and 1926 in the Netherlands with Brouwer preparing Urysohn’s unfinished work for publication. Their joint Mémoire sur les espaces topologiques compacts appeared in 1929.

Noether, Hopf and algebraic topology

From 1925 to 1932 Alexandrov spent every summer in Göttingen, taking part in Noether’s seminar and running a topology seminar of his own. In 1926 he met Heinz Hopf, and the two became inseparable friends and collaborators. Noether pressed both of them to see the Betti numbers of a space as invariants of homology groups, and Alexandrov later credited her influence on his own work on decompositions of spaces. With Hopf he spent 1927–28 in Princeton, working with Solomon Lefschetz, Oswald Veblen and James Alexander, and there the two planned a three-volume treatise on topology. Only the first volume, Topologie I, appeared, in 1935; the war ended the project.

His 1928 paper on the general concept of dimension introduced the nerve of a covering (Lesson 08). The idea connected the point-set topology of Moscow with the combinatorial topology of Princeton and Göttingen, and it is the basis of Čech homology. In 1929 he became professor at Moscow University and a corresponding member of the Academy of Sciences.

Borel sets and the A-operation

For the general reader. Georg Cantor asked whether every infinite set of real numbers is either countable or as large as the whole line. No one can answer this in general. Alexandrov proved that the answer is yes for all Borel sets, the sets built from intervals by countable unions, intersections and complements, which include every set an analyst meets in practice.

A set \(P\subseteq\mathbb R\) is perfect if it is closed, non-empty and has no isolated points. Every perfect set has cardinality \(2^{\aleph_0}\): inside it one can build a binary tree of disjoint closed intervals, giving a copy of the Cantor set.

The A-operation

Let \(\mathbb N^{<\mathbb N}\) be the set of finite sequences of positive integers. A Suslin scheme is a family \(\{F_s\}_{s\in\mathbb N^{<\mathbb N}}\) of sets. The A-operation produces

where \(x|n\) is the sequence of the first \(n\) terms of \(x\). Alexandrov showed that every Borel set is obtained by applying \(\mathcal A\) to a scheme of closed sets, and proved the perfect set property for every set so obtained. In 1917 Suslin, studying Alexandrov’s operation, found that it produces more than the Borel sets. He called the results analytic sets and proved that a set is Borel exactly when both it and its complement are analytic. The operation is called the A-operation in Russian texts and the Suslin operation elsewhere.

Proof sketch for analytic sets

An analytic set is a continuous image \(A=f(\mathbb N^{\mathbb N})\) of Baire space. For a finite sequence \(s\), let \(N_s\) be the set of \(x\) extending \(s\). Discard every \(N_s\) whose image is countable; countably many sets of countable image are discarded, so what remains still has uncountable image. In the remaining tree, every node has two extensions \(t,t'\) whose images contain points \(f(x)\ne f(x')\), and by continuity we can extend further so that the images lie in disjoint small closed intervals. Iterating gives a Cantor scheme whose branches map injectively and continuously onto a compact set without isolated points: a perfect subset of \(A\).

Compactness

For the general reader. A closed interval has a property the open interval lacks: a sequence inside it cannot escape, and any covering of it by open pieces can be cut down to finitely many. For metric spaces these properties agree. In general spaces they do not, and Alexandrov and Urysohn showed that the covering property is the one that matters.

A space \(X\) is compact (Alexandrov’s bicompact) if every open cover has a finite subcover. Alexandrov reserved “compact” for what is now called countably compact: every countable open cover has a finite subcover. A point \(x\) is a complete accumulation point of an infinite set \(A\subseteq X\) if \(|U\cap A|=|A|\) for every neighbourhood \(U\) of \(x\).

Proof sketch. (1)⇔(2) by taking complements. (1)⇒(3): if an infinite \(A\) had no complete accumulation point, every \(x\) would have an open \(U_x\) with \(|U_x\cap A|<|A|\); finitely many \(U_x\) cover \(X\), so \(A\) is a finite union of sets of smaller cardinality, impossible for infinite \(A\). (3)⇒(1): suppose an open cover has no finite subcover, and let \(\kappa\) be the least cardinality of a subcover. Well-order a subcover \(\{U_\alpha\}_{\alpha<\kappa}\). For each \(\alpha\), \(\bigcup_{\beta<\alpha}U_\beta\ne X\), so pick \(x_\alpha\) outside it. The set \(A=\{x_\alpha\}\) has cardinality \(\kappa\), yet each \(U_\beta\) contains only the \(x_\alpha\) with \(\alpha\le\beta\), fewer than \(\kappa\) points when \(\kappa\) is regular. A point of \(X\) lies in some \(U_\beta\), so it is not a complete accumulation point. The singular case needs a further cofinality argument.

In 1930 Andrey Tikhonov, Alexandrov’s student, proved that any product of closed intervals is compact, and in 1935 that any product of compact spaces is compact. This theorem, which is equivalent to the axiom of choice, shows why the open-cover definition is the right one: countable compactness is not preserved by products.

The one-point compactification

For the general reader. The plane becomes a sphere when one adds a single point at infinity; this is how complex analysts have long treated the Riemann sphere. In 1924 Alexandrov showed that the same trick works for any locally compact space and explained exactly when the result is well behaved.

A Hausdorff space \(X\) is locally compact if every point has a neighbourhood with compact closure. Let \(X^{*}=X\cup\{\infty\}\) and declare open: the open subsets of \(X\), and the sets \(\{\infty\}\cup(X\setminus K)\) for compact \(K\subseteq X\).

Proof sketch. The open sets of \(X^*\) form a topology because compact sets are closed in the Hausdorff space \(X\) and finite unions and arbitrary intersections of compact closed sets are compact. Given an open cover of \(X^*\), one member is \(\{\infty\}\cup(X\setminus K)\); finitely many others cover the compact \(K\). Two points of \(X\) are separated as in \(X\). A point \(x\in X\) is separated from \(\infty\) exactly when \(x\) has an open neighbourhood \(U\) inside a compact \(K\): take \(U\) and \(\{\infty\}\cup(X\setminus K)\). That is local compactness.

The construction appeared in Alexandrov’s 1924 paper on the metrization of locally compact spaces. It gives one-line proofs of many facts: for example, a locally compact Hausdorff space is completely regular, because it is a subspace of a compact Hausdorff space.

Metrization

For the general reader. When can the open sets of a space be described by a distance function? This metrization problem was posed by the Moscow school, and the first answer, a necessary and sufficient condition, was found by Alexandrov and Urysohn in 1923. Every later metrization theorem refines their idea of a sequence of ever finer coverings.

For a cover \(\mathcal U\) of \(X\) and \(x\in X\), the star of \(x\) is \(\mathrm{St}(x,\mathcal U)=\bigcup\{U\in\mathcal U:x\in U\}\).

Idea of sufficiency. Define \(\rho(x,y)=2^{-n}\) where \(n\) is the largest index for which \(x\) and \(y\) lie in a common member of \(\mathcal U_n\) (and \(\rho=0\) if \(x=y\)). Condition 1 gives a weak triangle inequality, \(\rho(x,z)\le2\max(\rho(x,y),\rho(y,z))\), and a standard chain construction, taking the infimum of \(\sum\rho(x_{i},x_{i+1})\) over finite chains from \(x\) to \(z\), produces a true metric equivalent to \(\rho\). Condition 2 makes its balls a base for the topology.

Paracompactness entered metrization through the Alexandrov–Urysohn circle of ideas. Arthur Stone proved in 1948 that every metric space is paracompact, and Yuri Smirnov, Alexandrov’s student, proved that a paracompact Hausdorff space that is locally metrizable is metrizable.

Nerves and Čech homology

For the general reader. Cover a shape with overlapping patches and record only which patches overlap. Alexandrov saw that this combinatorial record, the nerve, captures the shape’s holes when the patches are small and simple enough. Refining the cover and passing to the limit gives a homology theory for any compact space, however irregular.

Čech homology

If \(\mathcal V\) refines \(\mathcal U\), choosing for each \(V\in\mathcal V\) some \(U\supseteq V\) defines a simplicial map \(N(\mathcal V)\to N(\mathcal U)\), well defined up to contiguity, hence a well-defined map in homology. Finite open covers of a compact space, ordered by refinement, form a directed system, and

Alexandrov developed this inverse-limit approach between 1925 and 1929, describing a compact metric space as the limit of a sequence of polyhedra, the nerves of finer and finer covers (a projective spectrum). Eduard Čech gave the general form in 1932. Čech cohomology agrees with singular cohomology on polyhedra and manifolds but behaves better on wild spaces, and it is the cohomology of sheaf theory.

Dimension theory

For the general reader. A curve can be covered by small patches with at most two overlapping at any point; a surface needs three. Henri Lebesgue observed this, Brouwer proved it, and Urysohn and Karl Menger built a theory of dimension around such ideas. Alexandrov added two decisive tools: approximating a space by polyhedra through nerves, and measuring dimension with homology.

The order of a cover is the largest \(n\) such that some \(n+1\) members have a common point; equivalently, the dimension of its nerve. A normal space has covering dimension \(\dim X\le n\) if every finite open cover has a finite open refinement of order at most \(n\). Lebesgue’s covering theorem, proved by Brouwer, says \(\dim[0,1]^n=n\).

The maps are the canonical maps to nerves: given a cover \(\{U_i\}\) of small mesh and order \(n\), with a partition of unity \(\{\varphi_i\}\) subordinate to it, send \(x\) to the point with barycentric coordinates \((\varphi_i(x))\) in \(N(\mathcal U)\).

Homological dimension

For an abelian group \(G\), the cohomological dimension \(\dim_G X\) of a compact space is the largest \(n\) such that \(\check H^n(X,A;G)\ne0\) for some closed \(A\subseteq X\).

Alexandrov asked whether the finite-dimensionality hypothesis could be removed. Alexander Dranishnikov answered no in 1988 with an infinite-dimensional compactum of integral cohomological dimension 3. For other coefficient groups the two dimensions can differ even for finite-dimensional spaces: Pontryagin’s surfaces of 1930, built under Alexandrov’s influence, are 2-dimensional compacta whose product has dimension 3, showing that \(\dim(X\times Y)=\dim X+\dim Y\) can fail.

Alexandrov spaces and finite topology

For the general reader. Most topologies we meet are about continuity on the real line. Alexandrov noticed in 1937 that topology also makes sense on finite sets, and that there it is the same thing as an ordering. Decades later these finite spaces turned out to model every finite simplicial complex up to weak homotopy.

A topology is an Alexandrov topology if arbitrary intersections of open sets are open. Every topology on a finite set has this property. In such a space each point \(x\) has a smallest open neighbourhood \(U_x\), the intersection of all open sets containing it. Write \(x\le y\) if \(y\in U_x\), equivalently \(U_y\subseteq U_x\).

Proof sketch. Given a preorder, up-sets are closed under arbitrary unions and intersections, so they form an Alexandrov topology, and the smallest open set containing \(x\) is \(\uparrow x=\{y:y\ge x\}\). Conversely, an open set containing \(x\) contains \(U_x\), so it is an up-set; and \(U_x=\uparrow x\) is open, so every up-set, being a union of sets \(U_x\), is open. \(T_0\) says distinct points have distinct smallest neighbourhoods, which is antisymmetry.

The Moscow school, 1929–1953

From 1924 Alexandrov ran a topology seminar at Moscow University, and from 1929, as professor, he built it into a school. His doctoral students included Andrey Tikhonov, Lev Pontryagin, Aleksandr Kurosh, Lev Tumarkin and Yuri Smirnov. Kurosh came to algebra through lectures Alexandrov gave in Smolensk under Noether’s influence. Noether herself spent the winter of 1928–29 in Moscow, teaching abstract algebra at the university and leading a seminar on algebraic geometry at the Communist Academy. When she was dismissed in 1933, Alexandrov tried to secure her a chair in Moscow; the ministry delayed, and she went to Bryn Mawr. On 5 September 1935 he gave her memorial address to the Moscow Mathematical Society.

In 1929 his friendship with Andrey Kolmogorov began, on a long boat trip down the Volga. The two shared a home for the rest of their lives, from 1935 the house at Komarovka outside Moscow, where Jacques Hadamard, Maurice Fréchet, Stefan Banach, Kazimierz Kuratowski and Hopf were among the visitors. Students remembered skiing, swimming in the river and long walks alongside the mathematics.

He became president of the Moscow Mathematical Society in 1932 and held the post until 1964. In the summer of 1936, at the height of the purges, he was among Luzin’s former students who accused their teacher before an Academy commission of appropriating the work of students such as Suslin and Pyotr Novikov. Luzin lost his positions but was not arrested. The decision against him was formally reversed by the Russian Academy of Sciences only in 2012. Historians continue to debate the motives of the accusers.

During the war the Academy institutes were evacuated to Kazan, and Alexandrov’s own account dates his work on exact sequences and duality to that period. In 1943 he received the Stalin Prize, and in 1953 he became a full member of the Academy.

Later years and legacy, 1953–1982

In the 1950s and 1960s Alexandrov remained the leading figure of Soviet topology and one of the public faces of Soviet mathematics abroad. He spoke at the International Congress of Mathematicians in Amsterdam in 1954 and was vice-president of the International Mathematical Union from 1958 to 1962. For many years he edited Uspekhi Matematicheskikh Nauk. In 1955 he signed the Letter of Three Hundred against Trofim Lysenko’s domination of Soviet biology. In 1954 he began a seminar for first-year students, and he continued teaching undergraduates into old age.

He received the title Hero of Socialist Labour in 1969, the Lobachevsky Prize in 1972, several Orders of Lenin, and election to many foreign academies, including the Leopoldina, the American Philosophical Society and the US National Academy of Sciences. He was blind for the last years of his life. He died in Moscow on 16 November 1982. Weeks later Moscow University created a Department of General Topology and Geometry, headed by his student Vitaly Fedorchuk.

Legacy

• General topology: compactness, local compactness, the one-point compactification, the first metrization theorem and Alexandrov topologies are standard in every course.
• Algebraic topology: nerves of coverings and inverse limits of polyhedra are the foundation of Čech cohomology and of the passage from point-set to combinatorial topology.
• Dimension theory: polyhedral approximation and cohomological dimension, with the problem Dranishnikov solved in 1988.
• Books: Topologie I with Hopf (1935), Combinatorial Topology, and his introduction to set theory and the theory of functions.
• School: through Tikhonov, Pontryagin, Kurosh, Smirnov and their students, much of Soviet topology, algebra and control theory descends from his seminar.

Connections

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

Teachers and advisors

• Dmitri Egorov
• Nikolai Luzin 1914-1917

Students

• Aleksandr Kurosh
• Andrey Tikhonov
• Lev Pontryagin
• Lev Tumarkin
• Petru Soltan
• Vitaly Fedorchuk
• Yuri Smirnov

Influenced by

• Felix Hausdorff
• L. E. J. Brouwer 1924-1926

Influenced

• Eduard Čech

Colleagues and collaborators

• Heinz Hopf 1926-1935
• Solomon Lefschetz 1927-1928
• Mikhail Suslin 1915-1917
• Pavel Urysohn 1922-1924

Friends

• Andrey Kolmogorov 1929-1982
• Emmy Noether 1923-1935
• Heinz Hopf 1925-1971
• L. E. J. Brouwer

Also mentioned in this life

• André Weil
• Stefan Banach
• Émile Borel
• Georg Cantor
• Richard Courant
• Maurice Fréchet
• Jacques Hadamard
• David Hilbert
• Kazimierz Kuratowski
• Henri Lebesgue
• Nikolai Lobachevsky
• Bernhard Riemann

Fields

• general topology 3 lives
• algebraic topology 6 lives
• set theory 12 lives

On CoursesHub

• Homology & Cohomology · Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Pavel Sergeevich Aleksandrov, MacTutor History of Mathematics.
2. Aleksandrov, Pavel Sergeevich, Dictionary of Scientific Biography entry, via MacTutor.
3. Pavel Alexandrov, Wikipedia.
4. Александров Павел Сергеевич, Большая российская энциклопедия.
5. P. S. Alexandrov, In memory of Emmy Noether (address of 5 September 1935).
6. P. S. Alexandrov, Pages from an autobiography, Russian Math. Surveys 34:6 (1979) and 35:3 (1980).
7. Luzin affair, Wikipedia.

Mathematics

1. P. Alexandroff, Sur la puissance des ensembles mesurables B, C. R. Acad. Sci. Paris 162 (1916).
2. P. Alexandroff, Über die Metrisation der im Kleinen kompakten topologischen Räume, Math. Ann. 92 (1924) 294–301.
3. P. Alexandroff, Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, Math. Ann. 98 (1928) 617–635.
4. P. Alexandroff and P. Urysohn, Mémoire sur les espaces topologiques compacts, Verh. Kon. Akad. Wetensch. Amsterdam 14 (1929).
5. P. Alexandroff, Diskrete Räume, Mat. Sbornik 2 (1937).
6. P. Alexandroff and H. Hopf, Topologie I, Springer, 1935.
7. P. S. Alexandrov, Some results in the theory of topological spaces obtained within the last twenty-five years, Toposym 1961.
8. R. Engelking, General Topology, Heldermann, 1989.
9. M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966).
10. Aleksandrov compactification, Encyclopedia of Mathematics; Nerve complex and Alexandrov topology, Wikipedia.
11. Pavel Aleksandrov, nLab (references).