Generic objects in topology
| Thesis title in Czech: | Generické objekty v topologii |
|---|---|
| Thesis title in English: | Generic objects in topology |
| Key words: | generický objekt|Baireova věta|topologický prostor|kontinuum|dynamický systém |
| English key words: | generic object|Baire theorem|topological space|continuum|dynamical system |
| Academic year of topic announcement: | 2022/2023 |
| Thesis type: | dissertation |
| Thesis language: | angličtina |
| Department: | Department of Mathematical Analysis (32-KMA) |
| Supervisor: | doc. Mgr. Benjamin Vejnar, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 07.09.2023 |
| Date of assignment: | 07.09.2023 |
| Confirmed by Study dept. on: | 05.10.2023 |
| Guidelines |
| There is a rich variety of generic objects in topology. Let us mention only three of them: the Cantor set (generic in the hyperspace of all compact metric spaces), the pseudo-arc (generic in the hyperspace of continua), the special homeomorphism of the Cantor set (generic in the space of all Cantor set homeomorphisms with the uniform topology) [Kechris, Rosendal 2007]. Some of these objects are classical (Cantor set), some of them are well known, but stil of deep interest (pseudo-arc) and some of them are intensively studied (special homeomorphism of the Cantor set [Kupka, Oprocha 2019]).
The aim of this thesis is first to study the generic framework, which is covered by Classical descriptive set theory by Kechris, Continuum theory by Nadler and Topological and symbolic dynamics by Kurka. After that, some known concrete examples of generic objects (which occur in metrizable topology and classical dynamical systems) will be studied. Finally, the student will try to solve some open questions regarding these objects or he can try to observe some generic objects, which stayed hidden till now. |
| References |
| Nadler: Continuum theory
Kechris: Classical descriptive set theory Kechris, Rosendal: Turbulence, amalgamation, and generic automorphisms of homogeneous structures, 2007 Kupka, Oprocha: On the dynamics of generic maps on the Cantor set, 2019 Kurka: Topological and symbolic dynamics |