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2-manifolds and inverse limits of set-valued functions on intervals

  • * Corresponding author: Sina Greenwood

    * Corresponding author: Sina Greenwood 
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  • Suppose for each $n\in\mathbb{N}$, $f_n \colon [0,1] \to 2^{[0,1]}$ is a function whose graph $\Gamma(f_n) = \left\lbrace (x,y) \in [0,1]^2 \colon y \in f_n(x)\right\rbrace$ is closed in $[0,1]^2$ (here $2^{[0,1]}$ is the space of non-empty closed subsets of $[0,1]$). We show that the generalized inverse limit $\varprojlim (f_n) = \left\lbrace (x_n) \in [0,1]^\mathbb{N} \colon \forall n \in \mathbb{N},\ x_n \in f_n(x_{n+1})\right\rbrace$ of such a sequence of functions cannot be an arbitrary continuum, answering a long-standing open problem in the study of generalized inverse limits. In particular we show that if such an inverse limit is a 2-manifold then it is a torus and hence it is impossible to obtain a sphere.

    Mathematics Subject Classification: Primary: 54C08, 54E45; Secondary: 54F15, 54F65.

    Citation:

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  • Figure 1.  A torus as a GIL on intervals

    Figure 2.  A circle as a binary Mahavier product of simply-connected spaces

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