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The Lelek Fan as the Inverse Limit of Intervals with a Single Set-Valued Bonding Function Whose Graph is an Arc

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Abstract

We consider a family of inverse limits of inverse sequences of closed unit intervals with a single upper semi-continuous set-valued bonding function whose graph is an arc; the graph is the union of two line segments in \([0,1]^2\), both of which contain the origin (0, 0) and have positive slope. One of the segments extends to the top-boundary and the other to the right side boundary of \([0,1]\times [0,1]\). We show that there is a large subfamily \(\mathcal F\) of these bonding functions such that for each \(f\in \mathcal F\), the inverse limit of the inverse sequence of closed unit intervals using f as a single bonding function is homeomorphic to the Lelek fan.

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References

  1. Akin, E.: General Topology of Dynamical Systems, Volume 1, Graduate Studies in Mathematics Series, American Mathematical Society, Providence RI, (1993)

  2. Alvin, L., Kennedy, J., Kelly, J., Greenwood, S.: \(m\)-manifolds and inverse limits of set-valued functions on intervals, preprint (2022)

  3. Banič, I., Črepnjak, M., Merhar, M., Sovič, T., Milutinovič, U.: Ważewski’s universal dendrite as an inverse limit with one set-valued bonding function, Glas. Mat. 48 (2013) 137–165

  4. Banič, I., Martínez-de-la-Vega, V.: Universal dendrite \(D_3\) as a generalized inverse limit. Houston J. Math. 41, 669–682 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Charatonik, W.J.: The Lelek fan is unique. Houston J. Math. 15, 27–34 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Črepnjak, M., Sovič, T.: Standard universal dendrite \(D_m\) as an inverse limit with one set-valued bonding function. Topol. App. 209, 263–274 (2016)

    Article  MATH  Google Scholar 

  7. Greenwood, S., Suabedissen, R.: 2-manifolds and inverse limits of set-valued functions on intervals. Discrete Continuous Dyn Syst- Ser. A 37, 5693–5706 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Illanes, A.: A circle is not the generalized inverse limit of a subset of \([0,1]^{2}\). Proc. AMS 139, 2987–2993 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ingram, W.T.: An Introduction to Inverse Limits with Set-Valued Functions. Springer, New York (2012)

    Book  MATH  Google Scholar 

  10. Ingram, W.T., Mahavier, W.: Inverse Limits: From Continua to Chaos. Springer, New York, NY (2012)

    Book  MATH  Google Scholar 

  11. Lelek, A.: On plane dendroids and their end-points in the classical sense, Fund. Math. 49, 301–319 (1960/1961)

  12. Lemež, B.: An uncountable family of upper semicontinuous functions \(F\) such that the graph of \(F\) is homeomorphic to the inverse limit of closed unit intervals with \(F\) as the only bonding function. Topology App. 253, 25–37 (2019)

  13. Mahavier, W.S.: Inverse limits with subsets of \([0,1]\times [0,1]\). Topol. App. 141, 225–231 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nadler, S.B.: Continuum theory. An introduction, Marcel Dekker Inc, New York (1992)

  15. Nall, V.: The only finite graph that is an inverse limit with a set valued function on \([0,1]\) is an arc. Topol. App. 159, 733–736 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Overseegen, L., Bula, W.D.: A Characterization of smooth Cantor Bouquets, Proc. Amer.Math.Soc. 108, 529–534 (1990)

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Acknowledgements

The authors thank the anonymous referee for careful reading and constructive remarks that helped us improve the paper.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, 2000, Maribor, Slovenia

    Iztok Banič

  2. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Iztok Banič

  3. Andrej Marušič Institute, University of Primorska, Muzejski trg 2, 6000, Koper, Slovenia

    Iztok Banič

  4. Faculty of Science, University of Split, Rudera Boškovića 33, Split, Croatia

    Goran Erceg

  5. Lamar University, 200 Lucas Building, P.O. Box 10047, Beaumont, TX , 77710, USA

    Judy Kennedy

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  1. Iztok Banič
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  2. Goran Erceg
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  3. Judy Kennedy
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Correspondence to Iztok Banič.

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This work is supported in part by the Slovenian Research Agency (research project J1-4632 and research program P1-0285).

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Banič, I., Erceg, G. & Kennedy, J. The Lelek Fan as the Inverse Limit of Intervals with a Single Set-Valued Bonding Function Whose Graph is an Arc. Mediterr. J. Math. 20, 159 (2023). https://doi.org/10.1007/s00009-023-02323-3

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  • DOI: https://doi.org/10.1007/s00009-023-02323-3

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