Skip to main content
SpringerLink
Log in
Book cover

Shape in Picture pp 111–125Cite as

Shape Theory: An ANR-Sequence Approach

  • Conference paper

  • 185 Accesses

Part of the book series: NATO ASI Series ((NATO ASI F,volume 126))

Abstract

This is an expository article about the relationship of shape theory and some geometric notions. The ANR-sequence approach to shape theory is described. Then the shape classification of subcontinua of the plane is given. This is used to show that the dyadic solenoid is not the shape of any planar continuum. The shape classification of (m-sphere)-like continua is also obtained.

This is a preview of subscription content, 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   259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   329.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexandroff, P. (1929). Untersuchen über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension, Ann. of Math. 30, pp. 101–187.

    Article  MathSciNet  Google Scholar 

  2. Bing, R. H. (1960). A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math 12, pp. 209–230.

    MathSciNet  MATH  Google Scholar 

  3. Borsuk, K. (1968). Concerning homotopy properties of compacta, Fund. Math. 62, pp. 223–254.

    MathSciNet  MATH  Google Scholar 

  4. Borsuk, K. (1969/70). On movable compacta, Fund. Math. 66, pp. 137–146.

    Google Scholar 

  5. Eilenberg, S., Steenrod, N. (1952). Foundations of Algebraic Topology, Princeton University Press, Princeton.

    Google Scholar 

  6. Freudenthal, H. (1937). Entwicklungen von Räumen und ihren Gruppen, Compositio Math. 4, pp. 145–234.

    MathSciNet  MATH  Google Scholar 

  7. Kuratowski, K., (1933). Sur les transformations des sphères en des surfaces sphèriques, Fund. Math. 20, pp. 206–213.

    Google Scholar 

  8. Lefschetz, 5. (1931). On compact spaces, Ann. of Math. 32, pp. 521–538.

    Article  MathSciNet  Google Scholar 

  9. Lefschetz, S. (1942). Algebraic Topology. Am. Math. Soc., New York.

    Google Scholar 

  10. Mardešić, S., Segal, J. (1963). E-mappings onto polyhedra, Trans. Am. Math. Soc. 109, pp. 146–164.

    MATH  Google Scholar 

  11. Mardešić, S., Segal, J. (1966). A note on polyhedra embeddable in the plane, Duke Math. J. 33, pp. 633–638.

    MATH  Google Scholar 

  12. Mardešić, S., Segal, J. (1971). Shapes of compacta and ANR-systems, Fund. Math. 72, pp. 41–59.

    MATH  Google Scholar 

  13. Mardešić, S., Segal, J. (1982). Shape Theory, North-Holland, Amsterdam.

    MATH  Google Scholar 

  14. Patkowska, H. (1969). Some theorems on the embeddability of ANR-spaces into Euclidean spaces, Fund. Math. 65, pp. 289–308.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Jack Segal

Authors
  1. Jack Segal
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Centre for Mathematics and Computer Science (CWI), Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Ying-Lie O  & Henk J. A. M. Heijmans  & 

  2. Institute for Human Factors (TNO-IZF), Netherlands Organization for Applied Scientific Research, Kampweg 5, 3769 DE, Soesterberg, The Netherlands

    Alexander Toet

  3. Department of Communication and Neuroscience, University of Keele, ST5 5BG, Keele, Staffordshire, UK

    David Foster

  4. Department of Electrical and Computer Engineering, Rutgers University, 08855-0909, Piscataway, NJ, USA

    Peter Meer

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Segal, J. (1994). Shape Theory: An ANR-Sequence Approach. In: O, YL., Toet, A., Foster, D., Heijmans, H.J.A.M., Meer, P. (eds) Shape in Picture. NATO ASI Series, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03039-4_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-03039-4_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08188-0

  • Online ISBN: 978-3-662-03039-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation