Skip to main content
SpringerLink
Log in

An elementary approach to ‘Algebra ∩ topology = compactness’

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Herrlich and Strecker characterized the category Comp 2 of compact Hausdorff spaces as the only nontrivial full epireflective subcategory in the category Top 2 of all Hausdorff spaces that is concretely isomorphic to a variety in the sense of universal algebra including infinitary operations. The original proof of this result requires Noble's theorem, i.e. a space is compact Hausdorff iff every of its powers is normal, which is far from being elementary. Likewise, Petz' characterization of the class of compact Hausdorff spaces as the only nontrivial epireflective subcategory of Top 2, which is closed under dense extensions (= epimorphisms in Top 2) and strictly contained in Top 2 is based on a result by Katětov stating that a space is compact Hausdorff iff its every closed subspace is H-closed. This note offers an elementary approach for both, instead.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Herrlich, H. and Strecker, G. E.: Algebra ∩ Topology = Compactness, Gen. Top. Appl. 1 (1971), 283–287.

    Google Scholar 

  2. Herrlich, H. and Strecker, G. E.: Category Theory, 2nd edn, Heldermann, Berlin, 1982.

    Google Scholar 

  3. Katětov, M.: Über H-abgeschlossene und bikompakte Räume, Časopis Pěst. Mat. Fys. 69 (1940), 36–49.

    Google Scholar 

  4. Linton, F. E. J.: Some aspects of equational categories, in Proc. Conf. Categorical Algebra, Springer, Berlin, Heidelberg, 1966, pp. 84–94.

    Google Scholar 

  5. Noble, N.: Products with closed projections II, Trans. Amer. Math. Soc. 160 (1971), 169–183.

    Google Scholar 

  6. Petz, D.: A characterization of the class of compact Hausdorff spaces, Studia Sc. Math. Hung. 12 (1977), 407–408.

    Google Scholar 

  7. Richter, G.: A characterization of the Stone-Čech-compactification, in Categorical Topology, World Scientific, Singapore, 1989, pp. 462–476.

    Google Scholar 

  8. Richter, G.: Characterizations of algebraic and varietal categories of topological spaces, Topology Appl. 42 (1991), 109–125.

    Google Scholar 

  9. Richter, G.: Axiomatizing algebraically behaved categories of Hausdorff spaces, Canadian Mathematical Society Conference Proceedings 13 (1992), 367–389.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Universitätsstraße 25, D-33615, Bielefeld, Germany

    G. Richter

Authors
  1. G. Richter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Richter, G. An elementary approach to ‘Algebra ∩ topology = compactness’. Appl Categor Struct 4, 443–446 (1996). https://doi.org/10.1007/BF00122689

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00122689

Mathematics Subject Classifications (1991)

Key words

Search

Navigation