Relationship among various Vietoris-type and microsimplicial homology theories

Takuma Imamura

Archivum Mathematicum (2021)

  • Volume: 057, Issue: 3, page 131-150
  • ISSN: 0044-8753

Abstract

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In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi’s homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord’s homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi’s homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.

How to cite

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Imamura, Takuma. "Relationship among various Vietoris-type and microsimplicial homology theories." Archivum Mathematicum 057.3 (2021): 131-150. <http://eudml.org/doc/297620>.

@article{Imamura2021,
abstract = {In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi’s homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord’s homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi’s homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.},
author = {Imamura, Takuma},
journal = {Archivum Mathematicum},
keywords = {McCord homology; Korppi homology; $\mu $-homology; Vietoris homology; nonstandard analysis},
language = {eng},
number = {3},
pages = {131-150},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Relationship among various Vietoris-type and microsimplicial homology theories},
url = {http://eudml.org/doc/297620},
volume = {057},
year = {2021},
}

TY - JOUR
AU - Imamura, Takuma
TI - Relationship among various Vietoris-type and microsimplicial homology theories
JO - Archivum Mathematicum
PY - 2021
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 057
IS - 3
SP - 131
EP - 150
AB - In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi’s homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord’s homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi’s homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.
LA - eng
KW - McCord homology; Korppi homology; $\mu $-homology; Vietoris homology; nonstandard analysis
UR - http://eudml.org/doc/297620
ER -

References

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  8. Mardešić, S., Segal, J., Shape Theory, North-Holland Mathematical Library, vol. 26, North-Holland, Amsterdam-New York-Oxford, 1982. (1982) MR0676973
  9. McCord, M.C., 10.4064/fm-74-1-21-28, Fund. Math. 74 (1972), no. 1, 21–28. (1972) MR0300270DOI10.4064/fm-74-1-21-28
  10. Robinson, A., Non-standard Analysis, Studies in Logic and the Foundations of Mathematics, vol. 42, North-Holland, Amsterdam, 1966. (1966) MR0205854
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