Alexandrov's main works are on topology, set theory, theory of functions of a real variable, geometry, calculus of variations, mathematical logic, and foundations of mathematics. He introduced the new concept of
compactness (Alexandrov himself called it "Bicompactness", and applied the term compact to only countably compact spaces, as was customary before him). Together with P. S. Urysohn, Alexandrov showed the full meaning of this concept; in particular, he proved the first general
metrization theorem and the famous compactification theorem of any locally compact Hausdorff space by adding a single point. From 1923 P. S. Alexandrov began to study
combinatorial topology, and he managed to combine this branch of topology with general topology and significantly advance the resulting theory, which became the basis for modern
algebraic topology. It was he who introduced one of the basic concepts of algebraic topology — the concept of an
exact sequence. Alexandrov also introduced the notion of a
nerve of a covering, which led him (independently of E. Cech) to the discovery of Alexandrov-Cech Cohomology. In 1924, Alexandrov proved that in every open cover of a separable metric space, a locally finite open cover can be inscribed (this very concept, one of the key concepts in general topology, was first introduced by Alexandrov. In fact, this proved the paracompact nature of separable metric spaces (although the term "paracompact space" was introduced by
Jean Dieudonné in 1944, and in 1948
Arthur Harold Stone showed that the requirement of separability can be abandoned). He significantly advanced the theory of dimension. In particular, he founded the homological theory of dimension, defining its basic concepts in 1932. He developed methods of combinatorial research of general topological spaces, proving a number of basic laws of topological duality. In 1927, he generalized Alexander's theorem to the case of an arbitrary closed set. Alexandrov and
P. S. Urysohn were the founders of the Moscow topological school, which received international recognition. A number of concepts and theorems of topology bear Alexandrov's name: the Alexandrov compactification, the Alexandrov-Hausdorff theorem on the cardinality of a-sets, the Alexandrov topology, and the Alexandrov-Cech homology and cohomology. His books played an important role in the development of science and
mathematics education in Russia:
Introduction to the General Theory of Sets and Functions,
Combinatorial Topology,
Lectures on Analytical Geometry,
Dimension Theory (together with B. A. Pasynkov) and
Introduction to Homological Dimension Theory. The textbook
Topologie I, written together with
Heinz Hopf in German (Alexandroff P., Hopf H. (1935)
Topologie Band 1 — Berlin) became the classic course of topology of its time.
The Luzin Affair In 1936, Alexandrov was an active participant in the political offensive against his former mentor Luzin that is known as the
Luzin affair. Despite the fact that P. S. Alexandrov was a student of N. N. Luzin and one of the members of the , during the persecution of Luzin (the Luzin Affair), Alexandrov was one of the most active persecutors of the scientist. Relations between Luzin and Alexandrov remained very strained until the end of Luzin's life, and Alexandrov became an academician only after Luzin's death.
Students Among the students of P. S. Alexandrov, the most famous are
Lev Pontryagin,
Andrey Tychonoff and
Aleksandr Kurosh. The older generation of his students includes L. A. Tumarkin, V. V. Nemytsky, A. N. Cherkasov, N. B. Vedenisov, G. S. Chogoshvili. The group of "Forties" includes Yu. M. Smirnov, K. A. Sitnikov, O. V. Lokutsievsky, E. F. Mishchenko, M. R. Shura-Bura. The generation of the fifties includes A.V. Arkhangelsky, B. A. Pasynkov, V. I. Ponomarev, as well as E. G. Sklyarenko and A. A. Maltsev, who were in graduate school under Yu.M. Smirnov and K. A. Sitnikov, respectively. The group of the youngest students is formed by V. V. Fedorchuk, V. I. Zaitsev and E. V. Shchepin. ==Honours and awards==