**Etiuda Grant, Polish National Science Center**

Project: *GO-spaces and paracompactness in Cartesian products *

ETIUDA is a funding opportunity intended for PhD candidates. The duration of the scholarship is up to 12 months and it includes a 6-month internship in a research institute outside of Poland. In 2014, the National Science Center has awarded grants to the best 35 out of 105 applications in the fields of Physical Sciences and Engineering. During the internship, I worked at Bar-Ilan University in Israel under supervision of Professor Boaz Tsaban.

### About the project

Paracompactness is a common generalization of compactness and metrizability. One of the classic problems of general topology is to **find a characterization of the class of all productively paracompact spaces, that is, spaces whose product with every paracompact space is paracompact.**

In 1975, Ratislav Telgarsky introduced some infinite topological two player game. He proved that if the first player has a winning strategy in his game played on a paracompact space $X$, then the space $X$ is productively paracompact. In 2009, Kazimierz Alster **conjectured that a paracompact space $X$ is productively paracompact if and only if the first player has a winning strategy in Telgarsky's game played on $X$.**

Prior to this project, the above conjecture was verified positively for metrizable spaces, closed images of metrizable spaces, and some spaces with weight at most $\omega_1$.

Investigations of productively paracompact spaces are based on testing spaces. From a productively paracompact space $X$ a concrete space $Y$ is constructed such that the paracompactness of the product $X\times Y$ provides internal information on the space $X$. Construction of testing spaces follows from counterexamples for paracompactness in products. Prominent counterexamples are GO-spaces (generalized ordered spaces), i.e., linearly ordered spaces with a topology generated by intervals. Since it is very difficult to settle this problem in the general case, it was natural to restrict consideration to spaces related to GO-spaces. **The goal of this project was to characterize productively paracompact GO-spaces, productively paracompact closed images of GO-spaces, and consider other covering properties in products.**

In this project we proved that **the above conjecture is true for first-countable GO-spaces, GO-spaces with weight at most $\omega_1$, closed images of GO-spaces defined on the real line and some of closed images of GO-spaces defined on subspaces of the real line**. Moreover, we discovered interconnections between productively paracompact spaces and selection principles theory. Selection principles theory deals with covering properties, such as Hurewicz's and Menger's properties mentioned below. We proved that **every separable productively paracompact space has Hurewicz's property**.

The results of this part of the project extend substantially the list of known theorems about productively paracompact spaces and form a partial solution to the general problem. They give us better understanding of the behavior of paracompactness in products. The results for closed images of GO-spaces motivate further investigations.

### The internship in Israel

During the internship, Professor Boaz Tsaban and I considered another than paracompactness covering properties in products. A space $X$ is Menger if for every sequence of open covers $\mathcal{O}_1, \mathcal{O}_2, \ldots$ , there are finite sets $\mathcal{F}_1\subset\mathcal{O}_1, \mathcal{F}_2\subset\mathcal{O}_2,\ldots$ such that the family $\mathcal{F}_1\cup\mathcal{F}_2\cup\dots$ is a cover of $X$.

One of the most important problems in the field of selection principles is **whether, in ZFC, there is a Menger set $M\subseteq\mathbb{R}$ whose square $M\times M$ is not Menger**. In this project we found a partial solution: ** using a very general set-theoretic assumption, such a set exists**. Our assumption holds in most of the canonical models of set-theory. The proof method is new, and it has many applications for other selection principles.

### Popularization of the results

The above resluts were presented at international topological conferences in Italy, Ireland, and Czech Republic. They were also discussed on seminars in Poland and Israel. All these results are contained in the following papers.

- Productivity of paracompactness and closed images of real GO-spaces, Topology and its Applications 222 (2017), 254-273.
- Products of Menger spaces: a combinatorial approach (with B. Tsaban), Annals of Pure and Applied Logic 168 (2017), 1-18.
- Products of Menger spaces, II: general spaces (with B. Tsaban), submitted.