Skip to main content
SpringerLink
Log in

Transversals and finite topologies

  • Contributed Papers
  • Conference paper
  • First Online:
Combinatorial Mathematics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 686))

Abstract

Steffens [10] has shown that a family A of finite sets has a transversal if and only if the collection of all ‘critical subfamilies’ is a topology on A. In [6] these ‘transversal topologies’ have been characterized as well as families whose transversal topologies satisfy separation axioms. The purpose of this paper is to apply these results to enumerating certain finite topologies.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C.E. Aull, and W.J. Thron, "Separation axioms between T0 and T1", Indag. Math. 24 (1962), 26–37.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Harris, and L. Schoenfeld, "The number of idempotent elements in symmetric semigroups", J. Comb. Th. 3 (1967), 122–135.

    Article  MathSciNet  MATH  Google Scholar 

  3. M.J. Huebener, "Complementation in the lattice of regular topologies", Ph.D. dissertation, Univ. of Cincinnati, 1970.

    Google Scholar 

  4. J. Knopfmacher, "Note on finite topological spaces", J. Austral. Math. Soc. 9 (1969), 252–256.

    Article  MathSciNet  MATH  Google Scholar 

  5. R.E. Larson, and S.J. Andima, "The lattice of topologies: a survey", Rocky Mt. J. of Math. 5 (1975), 177–198.

    Article  MathSciNet  MATH  Google Scholar 

  6. R.A. Razen, "A characterization of transversal topologies", to appear.

    Google Scholar 

  7. J. Riordan, "Forests of labeled trees", J. Comb. Th. 5 (1968), 90–103.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Sharp, Jr., "Cardinality of finite topologies", J. Comb. Th. 5 (1968), 82–86.

    Article  MathSciNet  MATH  Google Scholar 

  9. R.P. Stanley, "On the number of open sets of finite topologies", J. Comb. Th. 10 (1971), 74–79.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Steffens, "Injektive Auswahlfunktionen", Schriften aus dem Gebiet der Angewandten Mathematik Nr. 2, Aachen 1972.

    Google Scholar 

  11. D. Stephen, "Topology on finite sets", Amer. Math. Monthly 75 (1968), 739–741.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of New South Wales, P.O. Box 1, 2033, Kensington, New South Wales, Australia

    R. A. Razen

Authors
  1. R. A. Razen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

D. A. Holton Jennifer Seberry

Rights and permissions

Reprints and permissions

Copyright information

© 1978 Springer-Verlag

About this paper

Cite this paper

Razen, R.A. (1978). Transversals and finite topologies. In: Holton, D.A., Seberry, J. (eds) Combinatorial Mathematics. Lecture Notes in Mathematics, vol 686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062539

Download citation

  • DOI: https://doi.org/10.1007/BFb0062539

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08953-7

  • Online ISBN: 978-3-540-35702-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation