Kazimierz Kuratowski - Research

Research

Kuratowski’s research mainly focused on abstract topological and metric structures. He implemented the closure axioms (known in the world as the Kuratowski closure axioms), which was fundamental for the development of topological space theory and irreducible continuum theory between two points. The most valuable results, which were obtained by Kazimierz Kuratowski after the war are those that concern the relationship between topology and analytic functions (theory), and also research in the field of cutting Euclidean spaces. Together with Ulam, who was Kuratowski’s most talented student during the Lwów Period, he introduced the concept of so-called quasi homeomorphism that opened up a new field in topological studies. Kuratowski’s research in the field of measure theory, including research with Banach, Tarski, was continued by many students. Moreover, with Alfred Tarski and Wacław Sierpiński he provided most of the theory concerning Polish spaces (that are indeed named after these mathematicians and their legacy). Knaster's and Kuratowski's brought a comprehensive and precise study on connected components theory. It was applied to issues such as cutting-plane, with the paradoxical examples of connected components.

Kuratowski proved the Kuratowski-Zorn lemma (often called just Zorn's lemma) in 1922 (Fundamenta Mathematicae, vol.3). This result has important connections to many basic theorems. Zorn gave its application in 1935 ("Bulletin of the American Mathematical Society", 41). Kuratowski implemented many concepts in set theory and topology. In many cases, Kuratowski established new terminology and symbolism. His contributions to mathematics include:

• a characterization of Hausdorff spaces which are now called Kuratowski closure axioms;
• proof of the Kuratowski-Zorn lemma;
• in Graph theory, the characterization of Planar graphs now known as Kuratowski's theorem;
• identification of the ordered pair (x,y) with the set {{x},{x,y}};
• introduction of the Tarski-Kuratowski algorithm;
• Kuratowski's closure-complement problem;
• Kuratowski's free set theorem;
• Kuratowski convergence of subsets of metric spaces;
• the Kuratowski, Ryll-Nardzewski measurable selection theorem;

Kuratowski’s post-war works were mainly focused on three strands:

• The development of homotopy in continuous functions.
• The construction of connected space theory in higher dimensions.
• The uniform depiction of cutting Euclidean spaces by any of its subsets, based on the properties of continuous transformations of these sets.