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The hyperspace of \(\mathcal{T} \)-closed subcontinua. (English) Zbl 1441.54023

Given a metric continuum \(X\), the hyperspace of subcontinua is denoted by \(C(X)\) and is endowed with the Hausdorff metric. The Jones’ set function \(\mathcal {T}\) is defined on subsets of \(X\) as follows. If \(A\subset X\), let \(\mathcal {T}(A)=\{p\in X:\) for each \(M\in C(X)\), with \(p\in int_{X}(M)\), we have \(M\cap A\neq \emptyset\}\).
A subcontinuum \(A\) of \(X\) is a \(\mathcal {T}\)-closed subcontinuum of \(X\) provided that \(\mathcal {T}(A)=A\). The paper under review introduces the study of the hyperspace of \(\mathcal {T}\)-closed subcontinua of \(X\), which is denoted by \(C_{\mathcal{T}}(X)\).
The paper starts with a series of illustrating examples. Besides showing some geometric flavor on this topic, the authors also show that the structure of \(C_{\mathcal{T}}(X)\) can have unexpected behavior.
The authors also study some topological properties of \(C_{\mathcal{T}}(X)\), like connectedness, compactness and density of \(C_{\mathcal{T}}(X)\) in \(C(X)\) on some specific families of continua such as products, cones, compactifications of the ray \([0,\infty)\), fans and curves of pseudo-arcs.
The following list of interesting problems, offered by the authors, shows that the structure of \(C_{\mathcal{T}}(X)\) is complex and worth being studied.
Problem 3.18. Given a compact metric space \(Y\), does there exist a continuum \(X\) such that \(C_{\mathcal{T}}(X)\) is homeomorphic to \(Y\)?
Problem 4.11. Let \(X\) be a continuum and let \(Y\) be a finite graph. Is \(C_{\mathcal{T}}(X\times Y)\) connected (arcwise connected)?
Problem 4.12. Let \(X\) be a compact metric space. If \(\mathrm{Sus}(X)\) denotes the suspension of \(X\), is \(C_{\mathcal{T}}(\mathrm{Sus}(X))\) connected (arcwise connected)?
Problem 4.13. Let \(X\) and \(Y\) be continua. If \(Y\) is contractible and locally connected, is \(C_{\mathcal{T}}(X\times Y)\) connected (arcwise connected)?
Problem 4.14. Let \(X\) be a continuum. If \(X\) is aposyndetic, is \(C_{\mathcal{T}}(X)\) connected (arcwise connected)?
Problem 6.6. Let \(X\) be a continuum. Does there exist a continuum \(Y\) such that \(C(X)\) is homeomorphic to \(C_{\mathcal{T}}(Y)\)?
Problem 6.7. Let \(X\) be a continuum for which \(C_{\mathcal{T}}(X)\) is a nondegenerate continuum. Is it true that \(\mathcal{T}\) is continuous on \(X\)?

MSC:

54F16 Hyperspaces of continua
54B20 Hyperspaces in general topology
54B05 Subspaces in general topology
54F15 Continua and generalizations
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References:

[1] Bellamy, D. P.; Fernández, L.; Macías, S., On \(\mathcal{T} \)-closed sets, Topol. Appl., 195, 209-225 (2015) · Zbl 1333.54022
[2] Burgess, C. E., Separation properties and n-indecomposable continua, Duke Math. J., 23, 595-599 (1956) · Zbl 0072.40406
[3] Jones, F. Burton, Concerning non-aposyndetic continua, Am. J. Math., 70, 403-413 (1948) · Zbl 0035.10904
[4] Camargo, J.; Macías, S.; Uzcátegui, C., On the image of Jones’s set function \(\mathcal{T} \), Colloq. Math., 153, 1-19 (2018)
[5] Camargo, J.; Uzcátegui, C., Continuity of Jones’s set function \(\mathcal{T} \), Proc. Am. Math. Soc., 145, 893-899 (2017) · Zbl 1356.54026
[6] Charatonik, J. J.; Charatonik, W. J., Images of the Cantor fan, Topol. Appl., 33, 163-172 (1989) · Zbl 0697.54019
[7] Illanes, A.; Nadler, S. B., Hyperspaces (1999), Marcel Dekker: Marcel Dekker New York · Zbl 0933.54009
[8] Lewis, W., Continuous curves of pseudo-arcs, Houst. J. Math., 11, 91-99 (1985) · Zbl 0577.54039
[9] Macías, S., Atomic maps and T-closed set, Topol. Proc., 50, 97-100 (2017) · Zbl 1365.54014
[10] Macías, S., Topics on Continua, Pure and Applied Mathematics Series, vol. 275 (2005), Chapman & Hall/CRC, Taylor & Francis Group: Chapman & Hall/CRC, Taylor & Francis Group Boca Raton, London, New York, Singapore · Zbl 1081.54002
[11] Nadler, S. B., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158 (1992), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0757.54009
[12] Nadler, S. B., Hyperspaces of Set, Monographs and Textbooks in Pure and Applied Math., vol. 49 (1978), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, Basel
[13] Read, D. R., Confluent and related mappings, Colloq. Math., 29, 233-239 (1974) · Zbl 0247.54010
[14] Swingle, P. M., Generalized indecomposable continua, Am. J. Math., 52, 647-658 (1930) · JFM 56.1135.03
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