Closed subsets of compact-like topological spaces

• Bardyla, Serhii ^[1] ; Ravsky, Alex ^[2]
  1. [1] University of Vienna

     University of Vienna

     Innere Stadt, Austria

     
  2. [2] Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Nat. Acad. Sciences of Ukraine
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 21, Nº. 2, 2020, págs. 201-214
• Idioma: inglés
• DOI: 10.4995/agt.2020.12258
• Enlaces
  • Texto completo
• Resumen

  • We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces ofcountably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.

• Referencias bibliográficas
  • L. Anderson, R. Hunter, and R. Koch, Some results on stability in semigroups, Trans. Amer. Math. Soc. 117 (1965), 521-529. https://doi.org/10.1090/S0002-9947-1965-0171869-7
  • T. Banakh, S. Bardyla and A. Ravsky, Embedding topological spaces into Hausdorff κ-bounded spaces, preprint, arXiv:1906.00185v3
  • T. Banakh, S. Bardyla and A. Ravsky, Embeddings into countably compact Hausdorff spaces, preprint, arXiv:1906.04541.
  • T. Banakh, S. Dimitrova and O. Gutik, Embedding the bicyclic semigroup into countably compact topological semigroups, Topology Appl. 157,...
  • S. Bardyla, Embedding of graph inverse semigroups into CLP-compact topological semigroups, Topology Appl. 272 (2020), 107058. https://doi.org/10.1016/j.topol.2020.107058
  • S. Bardyla and O. Gutik, On a semitopological polycyclic monoid, Algebra Discr. Math. 21, no. 2 (2016), 163-183.
  • D. Dikranjan and E. Giuli, $S(n)$-$theta$-closed spaces, Topology Appl. 28 (1988), 59-74. https://doi.org/10.1016/0166-8641(88)90036-3
  • D. Dikranjan and J. Pelant, Categories of topological spaces with sufficiently many sequentially closed spaces, Cahiers de Topologie et Geometrie...
  • A. Dow, J. R. Porter, R.M. Stephenson, Jr. and R. G. Woods, Spaces whose pseudocompact subspaces are closed subsets, Appl. Gen. Topol. 5,...
  • R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.
  • O. Gutik and K. Pavlyk, Topological semigroups of matrix units, Algebra Discr. Math. 3 (2005), 1-17. https://doi.org/10.1007/s00233-005-0530-0
  • O. Gutik, K. Pavlyk and A. Reiter, Topological semigroups of matrix units and countably compact Brandt $lambda^0$-extensions, Mat. Stud. 32,...
  • O. Gutik and A. Ravsky, On old and new classes of feebly compact spaces, Visn. Lviv. Univ. Ser. Mech. Math. 85 (2018), 48-59. https://doi.org/10.30970/vmm.2018.85.048-059
  • O. Gutik and D. Repovs, On countably compact 0-simple topological inverse semigroups, Semigroup Forum. 75, no. 2 (2007), 464-469. https://doi.org/10.1007/s00233-007-0706-x
  • J. Hildebrant and R. Koch, Swelling actions of $Gamma$-compact semigroups, Semigroup Forum 33, no. 1 (1986), 65-85. https://doi.org/10.1007/BF02573183
  • J. E. Joseph, On H-closed spaces, Proc. Amer. Math. Soc. 55, no. 1 (1976), 223-226. https://doi.org/10.2307/2041878
  • J. E. Joseph, More characterizations of H-closed spaces, Proc. Amer. Math. Soc. 63, no. 1 (1977), 160-164. https://doi.org/10.2307/2041087
  • D. D. Mooney, Spaces with unique Hausdorff extensions, Topology Appl. 61, no. 1 (1995), 241-256. https://doi.org/10.1016/0166-8641(94)00031-W
  • A. Osipov, Nearly H-closed spaces, Journal of Mathematical Sciences 155, no. 4 (2008), 626-633. https://doi.org/10.1007/s10958-008-9234-9
  • A. Osipov, Weakly H-closed spaces, Proceedings of the Steklov Institute of Mathematics 10, no. 1 (2004), S15-S17.
  • J. Porter, On locally H-closed spaces, Proc. London Math. Soc. 20, no. 3 (1970), 193-204. https://doi.org/10.1112/plms/s3-20.2.193
  • J. Porter and J. Thomas, On H-closed and minimal Hausdorff spaces, Trans. Amer. Math. Soc. 138 (1969), 159-170. https://doi.org/10.2307/1994905
  • J. Porter and C. Votaw, H-closed extensions I, Gen. Topology Appl. 3 (1973), 211-224. https://doi.org/10.1016/0016-660X(72)90013-X
  • J. Porter and C. Votaw, H-closed extensions II, Trans. Amer. Math. Soc. 202 (1975), 193-208. https://doi.org/10.2307/1997307
  • J. Porter and R. Woods, Extensions and Absolues of Hausdorff Spaces, Springer, Berlin, 1988, 856 pp. https://doi.org/10.1007/978-1-4612-3712-9
  • R. M. Stephenson, Jr, Initially κ-compact and related compact spaces, in: K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology,...
  • J.E. Vaughan, Countably compact and sequentially compact spaces, in: K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier,...
  • N. Velichko, H-Closed Topological Spaces, American Mathematical Society 78, no. 2 (1968), 103-118. https://doi.org/10.1090/trans2/078/05
  • J. Vermeer, Closed subspaces of H-closed spaces, Pacific Journal of Mathematics 118, no. 1 (1985), 229-247. https://doi.org/10.2140/pjm.1985.118.229