Explanation: Contrary to this statement, Pavel Alexandrov completed his university education at Moscow State University, where he studied under prominent mathematicians.

Explanation: Pavel Alexandrov's notable mentors during his education at Moscow State University were Dmitri Egorov and Nikolai Luzin, both distinguished mathematicians who guided his early academic development.

Explanation: While Alexandrov's mathematical work spanned several areas, his principal renown and foundational contributions were in set theory and topology, not primarily algebra and differential equations.

Explanation: Indeed, the Alexandroff compactification and the Alexandrov topology are significant concepts in the field of topology that bear Pavel Alexandrov's name, reflecting his foundational contributions.

Explanation: Alexandrov's mathematical contributions were not limited strictly to topology and set theory. His work also encompassed areas such as the theory of functions of a real variable, geometry, calculus of variations, and mathematical logic.

Explanation: Alexandrov introduced the term 'Bicompactness' to refer to spaces that were countably compact, differentiating this terminology from the broader use of 'compact' in topology.

Explanation: The Alexandrov-Urysohn theorem states that any locally compact Hausdorff space can be compactified by adding a single point, not two points. This is a fundamental result in the study of topological spaces.

Explanation: Indeed, Alexandrov's integration of combinatorial topology with general topology, beginning around 1923, was instrumental in laying the groundwork for the development of modern algebraic topology.

Explanation: The concept of an exact sequence in algebraic topology is attributed to Pavel Alexandrov, not Eduard Čech. This concept is a fundamental tool in the field.

Explanation: Alexandrov independently discovered the concept of the nerve of a covering, which led him to develop Alexandrov-Čech Cohomology. This work was done independently of Eduard Čech's contributions.

Explanation: In 1924, Alexandrov proved a significant result concerning separable metric spaces and open covers: it is possible to inscribe a locally finite open cover within any given open cover, not that they could not have such covers.

Explanation: Alexandrov's 1924 proof concerning inscribed open covers implicitly demonstrated the paracompact nature of separable metric spaces, a property later formally defined and named by Jean Dieudonné.

Explanation: In 1927, Alexandrov generalized Alexander's theorem, but not solely to open sets. His generalization extended the theorem's applicability to arbitrary closed sets, broadening its scope.

Explanation: Alexandrov's work on open covers did not demonstrate that separable metric spaces are always compact. Instead, his 1924 proof showed that such spaces possess the property of paracompactness.

Explanation: This is an accurate definition of the Alexandrov topology. It is a topological space where each point possesses a neighborhood base composed of open sets that include that point.

Explanation: The Alexandrov compactification is a method for extending a locally compact Hausdorff space to a compact space by adding a single point, often referred to as the 'point at infinity'.

Explanation: By merging combinatorial and general topology, Alexandrov laid the groundwork for modern algebraic topology, not abstract algebra. This integration was a significant advancement in the field.

Explanation: This is a correct characterization of the Alexandrov topology. It defines a topological space where each point has a neighborhood base consisting of open sets that include that point.

Explanation: This statement is accurate. Alexandrov's development of the concept of the nerve of a covering was foundational to his independent discovery of Alexandrov-Čech Cohomology.

Explanation: Pavel Alexandrov is primarily renowned for his seminal contributions to set theory and topology, which formed the core of his extensive mathematical work.

Explanation: Two significant topological concepts named in honor of Pavel Alexandrov are the Alexandroff compactification and the Alexandrov topology, reflecting his foundational work in the field.

Explanation: While Alexandrov's work extended to the theory of functions of a real variable, calculus of variations, and mathematical logic, differential geometry is not explicitly listed as a primary field of his contributions.

Explanation: Alexandrov introduced the term 'Bicompactness' to specifically denote spaces that are countably compact, thereby distinguishing it from the broader definition of compactness.

Explanation: The Alexandrov-Urysohn theorem asserts that any locally compact Hausdorff space can be compactified by the addition of precisely one point.

Explanation: Alexandrov's integration of combinatorial topology with general topology was highly significant, as it laid the foundational groundwork for the development of modern algebraic topology.

Explanation: The concept of an exact sequence, a fundamental tool in algebraic topology, is attributed to Pavel Alexandrov.

Explanation: Alexandrov's introduction of the concept of the 'nerve of a covering' was instrumental in his independent discovery of Alexandrov-Čech Cohomology.

Explanation: In 1924, Alexandrov proved that for any separable metric space, it is possible to inscribe a locally finite open cover within any given open cover. This is a key result in general topology.

Explanation: Alexandrov's 1924 proof concerning inscribed open covers implicitly demonstrated the paracompact nature of separable metric spaces.

Explanation: In 1927, Alexandrov generalized Alexander's theorem to apply not just to open sets, but to arbitrary closed sets, thereby extending its applicability.

Explanation: The Alexandrov compactification is primarily utilized as a method for extending locally compact Hausdorff spaces into compact spaces by the addition of a single point.

Explanation: Alexandrov's work on the nerve of a covering, which led to Alexandrov-Čech Cohomology, was discovered independently of Eduard Čech's contributions.

Explanation: Alexandrov did not view his work on the continuum problem as a success; rather, he considered it a 'serious disaster' and a personal setback, feeling unable to progress in mathematics at that time.

Explanation: This statement is accurate. Pavel Alexandrov is widely recognized as the founder of the homological theory of dimension, establishing its core concepts in 1932.

Explanation: This statement is correct. The Alexandrov-Hausdorff theorem, named in part after Pavel Alexandrov, is concerned with the cardinality of a-sets, contributing to set theory and topology.

Explanation: The book 'Introduction to the Theory of Dimension' was co-authored by Pavel Alexandrov and B. A. Pasynkov, not Pavel Urysohn.

Explanation: This is accurate. Pavel Alexandrov is credited with founding the homological theory of dimension, establishing its fundamental concepts and theorems in the early 1930s.

Explanation: This statement accurately describes the impact of Alexandrov's work on the continuum hypothesis. He experienced a significant personal crisis, feeling unable to progress in mathematics due to the difficulties encountered.

Explanation: The Alexandrov-Hausdorff theorem is not primarily concerned with differential geometry. Its focus is on the cardinality of a-sets, a topic within set theory and topology.

Explanation: Nikolai Luzin posed the challenge of determining the truth value of the continuum hypothesis to Alexandrov, a problem that led to a profound personal crisis and significantly impacted his mathematical trajectory.

Explanation: Alexandrov's reaction to the difficulties encountered with the continuum hypothesis problem was profound; he considered it a 'serious disaster' and a significant personal setback, feeling unable to progress in his mathematical endeavors.

Explanation: Pavel Alexandrov is credited as the founder of the homological theory of dimension, establishing its fundamental concepts and theorems in the early 1930s.

Explanation: The book 'Introduction to the Theory of Dimension' was co-authored by Pavel Alexandrov and B. A. Pasynkov, contributing significantly to the field of dimension theory.

Explanation: Pavel Alexandrov visited the University of Göttingen with his close collaborator Pavel Urysohn, but this occurred earlier, specifically in 1923 and 1924, not the early 1930s.

Explanation: While Nikolai Luzin did challenge Alexandrov with a significant problem that led to a creative crisis, the problem was related to the continuum hypothesis in set theory, not algebraic geometry.

Explanation: Ekaterina Romanovna Eiges, whom Alexandrov briefly married, was not a physicist. She was known as a poet, memoirist, librarian, and mathematician.

Explanation: The brief marriage of Alexandrov to Ekaterina Eiges ended not due to scientific differences, but because Alexandrov realized he was gay and that any marriage would be a personal mistake for him.

Explanation: This statement is accurate. Pavel Urysohn, a close collaborator and friend of Alexandrov, tragically drowned in a swimming accident in the Atlantic Ocean in August 1924.

Explanation: While Andrey Kolmogorov was a prominent mathematician and a close associate, he was not Alexandrov's student. They shared a lifelong partnership and a deep friendship, which Alexandrov characterized by mutual understanding and sympathy.

Explanation: The influential Moscow topological school was founded by Pavel Alexandrov and Pavel Urysohn, not Andrey Kolmogorov. Their collaboration established a significant center for topological research.

Explanation: While Alexandrov did co-author 'Topologie I' with Heinz Hopf, the book was originally published in German in 1935, not Russian.

Explanation: Pavel Alexandrov and Pavel Urysohn were indeed collaborators who met at Moscow State University. Their shared academic environment fostered a significant professional and personal relationship.

Explanation: Alexandrov's personal life did include a brief marriage and a lifelong partnership with Andrey Kolmogorov. This partnership was characterized by deep friendship and mutual understanding.

Explanation: Alexandrov and Pavel Urysohn visited the University of Göttingen during the years 1923 and 1924, a period marked by significant collaboration.

Explanation: Ekaterina Romanovna Eiges, whom Alexandrov briefly married, was recognized for her multifaceted talents as a poet, memoirist, librarian, and mathematician.

Explanation: The marriage between Alexandrov and Ekaterina Eiges was very brief because Alexandrov came to the realization that he was gay, concluding that any marriage would be a personal mistake for him.

Explanation: Pavel Urysohn, a close partner and collaborator of Alexandrov, tragically died in a swimming accident in the Atlantic Ocean while they were vacationing together.

Explanation: Alexandrov characterized his long-term relationship with Andrey Kolmogorov as a friendship celebrating fifty years, marked by profound mutual understanding and sympathy, without any significant quarrels or misunderstandings.

Explanation: The influential Moscow topological school is credited as being founded by Pavel Alexandrov and Pavel Urysohn, whose collaboration established a significant center for topological research.

Explanation: Alexandrov co-authored the influential topology textbook 'Topologie I' with Heinz Hopf. This seminal work was published in German in 1935.

Explanation: The influential textbook 'Topologie I,' co-authored by Alexandrov and Heinz Hopf, was originally published in German in 1935.

Explanation: Following his Ph.D. in 1927, Pavel Alexandrov did not move to the University of Kazan. Instead, he continued his distinguished academic career primarily at Moscow State University and the Steklov Institute of Mathematics.

Explanation: This statement is accurate. Pavel Alexandrov was elected as a member of the Russian Academy of Sciences in 1953, acknowledging his significant contributions to mathematics.

Explanation: Pavel Alexandrov is not buried in Novodevichy Cemetery. He was interred at the Kavezinsky cemetery, located in the Pushkinsky district of the Moscow region.

Explanation: Lev Pontryagin and Andrey Tychonoff are indeed mentioned as famous students of Pavel Alexandrov. Aleksandr Kurosh is also listed among his notable students.

Explanation: This statement is accurate. Pavel Alexandrov received numerous high Soviet honors, including the title Hero of Socialist Labour and the Stalin Prize, recognizing his significant scientific achievements.

Explanation: This statement is accurate. Pavel Alexandrov received international recognition for his work, including election as a member of the United States National Academy of Sciences in 1947.

Explanation: Alexandrov's 1961 book published by Dover Publications was titled 'Elementary concepts of topology,' not 'Advanced Concepts in Topology'.

Explanation: This statement is accurate. Alexandrov's autobiographical writings, titled 'Pages from an autobiography,' were published in 'Russian Mathematical Surveys' in 1979.

Explanation: This statement is accurate. Pavel Alexandrov was awarded the Order of Lenin an impressive six times, underscoring his distinguished service and achievements.

Explanation: Alexandrov's books, including 'Combinatorial Topology' and 'Introduction to the General Theory of Sets and Functions,' played a crucial role in advancing mathematics education and research within Russia.

Explanation: After obtaining his Ph.D. in 1927, Pavel Alexandrov primarily continued his academic work at Moscow State University and became affiliated with the Steklov Institute of Mathematics, solidifying his position in Soviet mathematics.

Explanation: Lev Pontryagin, Andrey Tychonoff, and Aleksandr Kurosh are listed as famous students of Pavel Alexandrov. Andrey Kolmogorov, while a close collaborator and partner, is not listed as his student.

Explanation: Pavel Alexandrov received the Order of Lenin multiple times, specifically six times, in recognition of his distinguished contributions and service.

Explanation: In 1946, Pavel Alexandrov received international academic recognition by being elected as a member of the American Philosophical Society.

Explanation: Alexandrov's 1961 book published by Dover Publications was titled 'Elementary concepts of topology,' making his foundational ideas accessible to a wider audience.

Explanation: Alexandrov's autobiographical writings, titled 'Pages from an autobiography,' were published in the journal 'Russian Mathematical Surveys' in 1979.

Explanation: This statement is accurate. In 1955, Alexandrov was among the signatories of the 'Letter of Three Hundred,' which publicly criticized the pseudoscientific movement known as Lysenkoism.

Explanation: Contrary to playing a supportive role, Alexandrov was actively involved in the persecution of his former mentor, Nikolai Luzin, during the 1936 Luzin Affair. He became one of Luzin's most vocal accusers.

Explanation: This statement is false. The 'Letter of Three Hundred,' signed by Alexandrov in 1955, was a critical document opposing Lysenkoism, not supporting it.

Explanation: In 1955, Alexandrov signed the 'Letter of Three Hundred,' a public statement that criticized Lysenkoism, a pseudoscientific doctrine that had gained considerable influence in Soviet biology and genetics.

Explanation: During the 1936 Luzin Affair, Alexandrov played a significant and negative role, acting as an active participant in the persecution of his former mentor, Nikolai Luzin.