Abstract
Let X be a continuum. By Remark 2.1.5, the image of any subset of X under \(\mathcal {T}\) is a closed subset of X. Then we may restrict the domain of \(\mathcal {T}\) to the hyperspace, 2X, of nonempty closed subsets of X. Since 2X has a topology, we may ask if \(\mathcal {T}\colon 2^{X}\to 2^{X}\) is continuous. The answer to this question is negative, as can be seen from Example 2.1.15. On the other hand, by Theorems 2.1.37 and 2.1.44, \(\mathcal {T}\) is continuous for locally connected continua and for indecomposable continua, respectively. In this chapter we present results related to the continuity of \(\mathcal {T}\) and examples of classes of decomposable nonlocally connected metric continua for which \(\mathcal {T}\) is continuous. In particular, we show that if a continuum X is almost connected im kleinen at each of its points and \(\mathcal {T}\) is continuous, then X is locally connected.
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Macías, S. (2021). Continuity of \(\mathcal {T}\). In: Set Function T. Developments in Mathematics, vol 67. Springer, Cham. https://doi.org/10.1007/978-3-030-65081-0_5
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DOI: https://doi.org/10.1007/978-3-030-65081-0_5
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