A study of universal elements in classes of bases of topological spaces

Dimitris N. Georgiou; Athanasios C. Megaritis; Inderasan Naidoo; Fotini Sereti

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 4, page 491-506
  • ISSN: 0010-2628

Abstract

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The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and studied, which is called quasi covering dimension and is denoted by dim q . In this paper, we define the base dimension-like function of the type dim q , denoted by b - dim q I F , and study the property of universality for this function. Especially, based on the method of “Universal spaces and mappings” by S. D. Iliadis (2005), we prove that in classes of bases which are determined by b - dim q I F there exist universal elements.

How to cite

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Georgiou, Dimitris N., et al. "A study of universal elements in classes of bases of topological spaces." Commentationes Mathematicae Universitatis Carolinae 62.4 (2021): 491-506. <http://eudml.org/doc/297903>.

@article{Georgiou2021,
abstract = {The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and studied, which is called quasi covering dimension and is denoted by $\dim _\{q\}$. In this paper, we define the base dimension-like function of the type dim$_\{q\}$, denoted by b - dim$^\{\rm I F\}_\{q\}$, and study the property of universality for this function. Especially, based on the method of “Universal spaces and mappings” by S. D. Iliadis (2005), we prove that in classes of bases which are determined by b - dim$^\{\rm I F\}_\{q\}$ there exist universal elements.},
author = {Georgiou, Dimitris N., Megaritis, Athanasios C., Naidoo, Inderasan, Sereti, Fotini},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological dimension; universality property; quasi covering dimension},
language = {eng},
number = {4},
pages = {491-506},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A study of universal elements in classes of bases of topological spaces},
url = {http://eudml.org/doc/297903},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Georgiou, Dimitris N.
AU - Megaritis, Athanasios C.
AU - Naidoo, Inderasan
AU - Sereti, Fotini
TI - A study of universal elements in classes of bases of topological spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 4
SP - 491
EP - 506
AB - The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and studied, which is called quasi covering dimension and is denoted by $\dim _{q}$. In this paper, we define the base dimension-like function of the type dim$_{q}$, denoted by b - dim$^{\rm I F}_{q}$, and study the property of universality for this function. Especially, based on the method of “Universal spaces and mappings” by S. D. Iliadis (2005), we prove that in classes of bases which are determined by b - dim$^{\rm I F}_{q}$ there exist universal elements.
LA - eng
KW - topological dimension; universality property; quasi covering dimension
UR - http://eudml.org/doc/297903
ER -

References

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  1. Engelking R., Theory of Dimensions Finite and Infinite, Sigma Series in Pure Mathematics, 10, Heldermann Verlag, Lemgo, 1995. Zbl0872.54002MR1363947
  2. Georgiou D. N., Megaritis A. C., Covering dimension and finite spaces, Appl. Math. Comput. 218 (2011), no. 7, 3122–3130. MR2851414
  3. Georgiou D. N., Megaritis A. C., An algorithm of polynomial order for computing the covering dimension of a finite space, Appl. Math. Comput. 231 (2014), 276–283. MR3174030
  4. Georgiou D. N., Megaritis A. C., Sereti F., A study of the quasi covering dimension for finite spaces through the matrix theory, Hacet. J. Math. Stat. 46 (2017), no. 1, 111–125. MR3585619
  5. Georgiou D. N., Megaritis A. C., Sereti F., A topological dimension greater than or equal to the classical covering dimension, Houston J. Math. 43 (2017), no. 1, 283–298. MR3647946
  6. Georgiou D. N., Megaritis A. C., Sereti F., 10.1016/j.topol.2020.107201, Topology Appl. 281 (2020), 107201, 11 pages. MR4174601DOI10.1016/j.topol.2020.107201
  7. Iliadis S., 10.1016/S0166-8641(00)90095-6, Topology Appl. 107 (2000), no. 1–2, 97–116. MR1783837DOI10.1016/S0166-8641(00)90095-6
  8. Iliadis S. D., Universal Spaces and Mappings, North-Holland Mathematics Studies, 198, Elsevier, Amsterdam, 2005. MR2126150
  9. Nagami K., Dimension Theory, Pure and Applied Mathematics, 37, Academic Press, New York, 1970. Zbl0224.54060MR0271918
  10. Nagata J., Modern Dimension Theory, Sigma Series in Pure Mathematics, 2, Heldermann Verlag, Berlin, 1983. Zbl0518.54002MR0715431
  11. Pears A. R., Dimension Theory of General Spaces, Cambridge University Press, Cambridge, 1975. Zbl0312.54001MR0394604

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