I was involved in
solving the following problems

  1. Is every countable Polish semigroup topologically isomorphic to a subsemigroup of IN^IN?

    2. This problem was posed in

    L. Elliott, J. Jonusas, Z. Mesyan, J. D. Mitchell, M. Morayne and Y. Peresse: "Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones", Trans. Amer. Math. Soc. 376 (2023), 8023--8093.

    A negative answer is published in

    S. Bardyla, L. Elliott, J. D. Mitchell and Y. Péresse: "Topological embeddings into transformation monoids", Forum Mathematicum (2024), published online.

  1. Does there exist a regular (first countable, separable) countably compact space on which every continuous real-valued function is constant?
  2. Does there exist, for every Hausdorff space R, a regular (first countable, separable) countably compact space on which every continuous function into R is constant?

    3. These problems were posed by Tzannes in

    E. Pearl: "Problems from Topology Proceedings", Topology Atlas, Toronto, 2003.

    The affirmative ZFC answer to both problems without properties in parentheses is published in

    S. Bardyla, A. Osipov: "On regular \kappa-bounded spaces admitting only constant continuous mappings into T_1 spaces of pseudo-character \leq \kappa", Act. Math. Hung. 163 (2021), 323--333.

    Consistent solutions of these problems with additional nice properties can be found in

    S. Bardyla, L. Zdomskyy: "On regular separable countably compact R -rigid spaces", Israel Journal of Mathematics 255 (2023), 783--810.

  1. Is every δ-set also a γ-set?
  2. Is every π-set also a δ-set?

    3. The first problem was originally posed by Gerlits and Nagy in

    J. Gerlits, Zs. Nagy: "Some properties of Cp(X), I", Topology Appl. 14 (1982), 151--161.

    Later it was reposed by Orenshtein and Tsaban in

    T. Orenshtein, B. Tsaban: "Pointwise convergence of partial functions: The Gerlits–Nagy Problem", Advances in Mathematics 232 (2013), 311--326.

    The second problem was posed by Sakai in

    M. Sakai: "Special subsets of reals characterizing local properties of function spaces", in: Koˇcinac L. (eds), Selection Principles and Covering Properties in Topology, quaderni di matematica 18, Caserta, 2006, 196--225.

    Affirmative consistent solutions to both problems are published in

    S. Bardyla, J. Šupina, L. Zdomskyy: "Ideal approach to convergence functional spaces", Trans. Amer. Math. Soc. 376 (2023), 8495--8528.

  1. Is it true that a Raikov complete topological group is H-closed in the class of quasitopological groups?

    2. This problem was posed by Arhangelskii and Choban in

    A. Arhangelskii, M. Choban: "Completeness type properties of semitopological groups, and the theorems of Montgomery and Ellis", Topology Proceedings 37 (2011), 33--60.

    The affirmative answer is published in

    S. Bardyla, O. Gutik, A. Ravsky: "H-closed quasitopological groups", Topology Appl. 217 (2017), 51--58.

  1. Is there a space with exactly three H-extensions? Four? Other non-Bell numbers?
  2. Is there a space with two one-point H-extensions and no other H-extensions?

    3. These problems were posed by Mooney in

    D. Mooney: "Spaces with unique Hausdorff extensions", Topology Appl. 61 (1995), 241–256.

    The affirmative answer to both problems is published in

    S. Bardyla, J. Supina, L. Zdomskyy: "Open filters and measurable cardinals", preprint, arXiv:2301.08704.

  1. Is it true that each locally compact topological graph inverse semigroup is discrete?

    2. This problem was posed by Mesyan, Mitchell, Morayne and Péresse in

    Z. Mesyan, J. D. Mitchell, M. Morayne, and Y. H. Péresse: "Topological graph inverse semigroups", Topology Appl. 208 (2016), 106--126.

    The characterization of graph inverse semigroups which admit only the discrete semigroup topology is published in

    S. Bardyla: "On locally compact topological graph inverse semigroups", Topology Appl. 267 (2019), 106873.

  1. Is it true that each Hausdorff locally compact monothetic topological monoid is a compact topological group?

    2. This problem was posed by Koch in

    R. Koch: "On monothetic semigroups", Proc. Amer. Math. Soc. 8 (1957), 397--401.

    It took more that 50 years until Zelenyuk constructed a counterexample to the above problem. We proved that a Hausdorff locally compact monothetic topological monoid S is a compact topological group if and only if S can be embedded into a Hausdorff quasitopological group. This result can be found in

    T. Banakh, S. Bardyla, I. Guran, O. Gutik, A. Ravsky: "Positive answers to Koch's problem in special cases", Topol. Alg. Appl. 8:1 (2020), 76--87.

  1. Is the countable pracompactness of paratopological groups preserved by Tychonoff products?

    2. This problem was posed by Ravsky in 2010 in the first version of the following paper

    T. Banakh, A. Ravsky: "On feebly compact paratopological groups", Topology Appl. 284 (2020), 107363.

    In 2018 Garcia-Ferreira and Tomita under CH provided a counterexample to the above problem. Assuming MA we constructed a Boolean countably compact topological group whose square is not countably pracompact. This example can be found in

    S. Bardyla, A. Ravsky, L. Zdomskyy: "A countably compact topological group with the non-countably pracompact square", Topology Appl. 279 (2020), 107251.

  1. Does there exists a pseudocompact topological semigroup which contains a closed copy of the bicyclic monoid?

    2. This problem was posed by Banakh, Dimitrova, and Gutik in

    T. Banakh, S. Dimitrova, O. Gutik: "Embedding the bicyclic semigroup into countably compact topological semigroups", Topology Appl. 157(18) (2010), 2803--2814.

    The affirmative answer is published in

    S. Bardyla, A. Ravsky: "Closed subsets of compact-like topological spaces", Applied General Topology, 21:2 (2020), 201--214.

  1. Is there an H-closed topological semilattice which is not absolutely H-closed?

    2. This problem was posed by Stepp in

    J.W. Stepp: "Algebraic maximal semilattices", Pacific J. Math. 58:1 (1975), 243--248.

    The affirmative answer is published in

    S. Bardyla, O. Gutik: "On H-complete topological semilattices", Mat. Stud. 38:2 (2012), 118--123.

  1. Is there an H-closed pospace which is not directed complete?

    2. This problem was posed by Yokoyama in

    T. Yokoyama: "On the relation between completeness and H-closedness of pospaces without infinite antichains", Alg. Discr. Math. 15:2 (2013), 287--294.

    The affirmative answer is published in

    T. Banakh, S. Bardyla: "Completeness and absolute H-closedness of topological semilattices", Topology Appl. 260 (2019) 189--202.