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A characterisation of the dimension of inverse limits of set-valued functions on intervals

Volume 249 / 2020

Iztok Banič, Sina Greenwood Fundamenta Mathematicae 249 (2020), 1-19 MSC: 54F15, 54C60, 54E45, 54H99. DOI: 10.4064/fm708-5-2019 Published online: 17 October 2019


For a sequence $\boldsymbol f$ of upper semicontinuous set-valued functions $f_i:\mathbb I _i\to 2^{\mathbb I _{i-1}}$ (where $\mathbb I _i=[0,1]$ for each $i\in \mathbb N $), we introduce the notion of a weighted sequence. We show that $\dim (\underleftarrow{\lim}\,{\boldsymbol f} )= n$ if and only if the maximal length of a weighted sequence admitted by $\boldsymbol f$ is $2n-2$. Furthermore, $\dim (\underleftarrow{\lim}\,{\boldsymbol f} )\ge n$ if and only if there is an increasing sequence $\langle i_1,\ldots ,i_n\rangle $ such that the projection of $\underleftarrow{\lim}\,{\boldsymbol f} $ to $\prod _{1\le j\le n}\mathbb I _{i_j}$ contains an $n$-cell.


  • Iztok BaničFaculty of Natural Sciences and Mathematics
    University of Maribor
    Koroška 160
    SI-2000 Maribor, Slovenia
  • Sina GreenwoodUniversity of Auckland
    Private Bag 92019
    Auckland, New Zealand

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