Projects / Programmes
General topology and set-theoretic methods
| Code |
Science |
Field |
Subfield |
| 1.01.02 |
Natural sciences and mathematics |
Mathematics |
Topology |
| Code |
Science |
Field |
| P110 |
Natural sciences and mathematics |
Mathematical logic, set theory, combinatories |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
Lindelöf space, D-space, Michael space, paracompact space, set-valued map, continuous selection, forcing, elementary submodel, cardinal characteristic of the continuum, selection principle
Researchers (17)
Organisations (1)
Abstract
This is a proposal for a 3-year joint Austrian-Slovenian research project, by means of which we wish to build the foundations for a leading international center for set-theoretic topology and mathematical logic in Slovenia, which will closely collaborate with the excellent Austrian Kurt Gödel research center for mathematical logic in Vienna (which is our equal partner in the proposed project) and other such leading centers around the world, in particular from European Union, Unites States, Russia, Japan and Israel.
The main theme of the proposed research is the application of set-theoretic methods such as forcing, elementary submodels, cardinal characteristics, and their syntheses with topological methods like selections of set-valued maps, to questions in general topology. For instance, we plan to investigate whether regular Lindelöf spaces and, more generally, paracompact spaces are D-spaces by using elementary submodels and set-valued maps. We also expect a significant progress in understanding productively Lindelöf spaces in various models of set theory and their relations to D-spaces, which may have a strong impact on the study of Michael spaces. In addition to solutions of some of the problems mentioned below we plan to contribute to a deeper understanding of the above-mentioned methods.
The notion of a D-space was introduced by van Douwen and Pfeffer in 1979 and since then it has been one of the main concepts in general topology. As it was noticed by Eisworth in 2007, one of the main objectives of the study of D-spaces is to gain a better understanding of the relationship between covering properties and the state of being a D-space. In particular, the following problems are open: Is every regular Lindelöf space X a D-space? Is every paracompact space a D-space?
First remarkable progress in this direction has recently been made by L. Aurichi, who proved that every space with Menger’s covering property is a D-space. In a joint work of Repovš and Zdomskyy we isolated and extended the combinatorial core of the Aurichi’s proof and established that under Martin’s Axiom, every paracompact space of size omega_1 is a D-space, thus solving a problem of G. Gruenhage. One of the aims of our project is to strengthen the abovementioned result, the ultimate goal being to avoid the additional set-theoretic assumptions, i.e., to answer the second problem mentioned above. Besides the use of appropriate games as before, we plan to use the theory of set-valued maps and their selections, where D. Repovš has an outstanding expertise. This approach is based on our observation that there are many natural set-valued maps with strong topological properties related to a neighborhood assignment N capturing all information about it. We expect that elementary submodels could be very useful here.
Another related problem is: Is there a Lindelöf space X whose product with the space of irrationals is not Lindelöf? Spaces X giving the affirmative solutions to this problem are called Michael spaces. There are many consistent examples of Michael spaces. However, the complete solution of this problem seems to be beyond the reach of currently known methods, and hence it is natural to approach it in “small” steps. One of the natural approaches implicitly suggested by F. Tall in his recent papers is to analyze productively Lindelöf sets of reals. It has been recently proved by Repovš and Zdomskyy that if there exists a Michael space, then every productively Lindelöf space has Menger’s property, and therefore is a D-space. This result seems to unveil a previously hidden connection between problems mentioned above, and we plan to explore it further.
Another intriguing area is the behavior of productively Lindelöf sets of reals under the negation of the Continuum Hypothesis. We plan to check whether all such sets are sigma-compact in the Miller and Sacks models of set theory. We expect affirmative answer for projective sets of reals.
Significance for science
This research project investigated several key important topics in modern set-theoretic topology and set theory. We have found new methods and techniques for resolving several open problems which have been in the center of attention by many leading experts for a long time. Therefore our results will definitely have impact on this field of mathematical sciences and will also support the progress of mathematical sciences in Slovenia. We have also discovered new ways to apply our results in other areas. We published our results in international mathematical journals which are placed high on the SCI journals list, e.g. Abstract and Applied Analysis, Advances in Mathematics, Archive for Mathematical Logic, Communications in Contemporary Mathematics, Communications on Pure and Applied Analysis, Computers & Mathematics with Applications, Forum Mathematicum, Geometry & Topology, Journal of Mathematical Analysis and Applications, Journal of Symbolic Logic, Linear Algebra and its Applications, Milan Journal of Mathematics, Proceedings of the Royal Society, Results in Mathematics, Transactions of the American Mathematical Society, etc. Our results received a lot of interest from the international mathematical community and have been cited many times. Some of our publications with the Elsevier North-Holland publishers were also among the most downloaded (e.g. in Topology and Its Applications). The project group is already very well established in its research area and it has already received many domestic and foreign awards. Members of our group have received numerous invitations to give lectures at important international conferences, confirming the international recognition of our research group. We had increased interest of foreign research institution for cooperation with our institute, especially from European Union. As a result, our research group has the largest number of international projects in mathematics.
Significance for the country
The work on this project has had a very positive influence on development of mathematical research in Slovenia, with the emphasis on general topology and set theory, and on its connection to the research networks worldwide, in particular to the European Union. The main results of this project is a discovery of very important new fundamental laws and their applications in mathematics, and extending the available research tools and their applications. Our results fit in very well with the plans for development of Slovenian science and technology, in the field of expansion of knowledge, as well as for a substantial improvement of the quality of the doctoral program. Our research is related to and builds upon past successful research in this field and is connected with the problems which have been very successfully studied, with a very positive feedback, in numerous international projects of our research. As the result of our longstanding efforts our institute is an internationally renowned European center of topology and one of the important meeting points of experts in this area. Our research has received several national and international prizes and we have been selected among the best program teams in the country. Several members of the group are already very influential internationally in their field of expertise. Our younger researchers, working within our group, have also very successfully began to establish themselves. We are very successfully cooperating with other sectors, e.g. we have developed new effective algorithms for generating discrete Morse functions in computational topology, which can be applied in medical radiological diagnostics. In this areas we are cooperating with some cutting-edge domestic hi-tech companies. Therefore we plan such productive collaboration also in the future. We shall also shall significantly expand our work on applied aspects of topology and strengthen our position in the EU research network. The project had also an extraordinary positive effect on the development of graduate studies in Slovenia, in particular the PhD programs in mathematics at the University of Ljubljana. Under our mentorship, our young researchers prepared their theses on the most up-to-date topics in topology and its applications. We have prepared modern graduate course like »Topology in computer science« at the Faculty of Computer Science and Informatics at the University of Ljubljana, which was of interest also for other fields, in particular in medicine.
Most important scientific results
Annual report
2013,
2014,
2015,
final report
Most important socioeconomically and culturally relevant results
Annual report
2013,
2014,
2015,
final report
