Butterfly points and hyperspace selections
Submitted: 2024-03-06
|Accepted:
|Published: 2024-10-01
Copyright (c) 2024 Valentin Gutev
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Keywords:
Vietoris topology, continuous selection, cut point, butterfy point
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Abstract:
If f is a continuous selection for the Vietoris hyperspace ℱ(X) of the nonempty closed subsets of a space X, then the point f(X)∊ X is not as arbitrary as it might seem at first glance. In this paper, we will characterise these points by local properties at them. Briefly, we will show that p=f(X) is a strong butterfly point precisely when it has a countable clopen base in cl(U) for some open set U⊂ X\{p} with cl(U)=U ∪ {p}. Moreover, the same is valid when X is totally disconnected at p=f(X) and p is only assumed to be a butterfly point. This gives the complete affirmative solution to a question raised previously by the author. Finally, when p=f(X) lacks the above local base-like property, we will show that ℱ(X) has a continuous selection h with the stronger property that h(S)=p for every closed S⊂X with p∈S.
References:
D. Bertacchi and C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl. 88 (1998), 179-197. https://doi.org/10.1016/S0166-8641(97)00175-2
H. Delfs and M. Knebusch, Locally semialgebraic spaces, Lecture Notes in Mathematics, vol. 1173, Springer-Verlag, Berlin, 1985. https://doi.org/10.1007/BFb0074551
A. Dow and S. Shelah, Tie-points and fixed-points in $mathbb{N}^*$, Topology Appl. 155 (2008), no. 15, 1661-1671. https://doi.org/10.1016/j.topol.2008.05.002
S. Eilenberg, Ordered topological spaces, Amer. J. Math. 63 (1941), 39-45. https://doi.org/10.2307/2371274
S. García-Ferreira, V. Gutev, T. Nogura, M. Sanchis, and A. Tomita, Extreme selections for hyperspaces of topological spaces, Topology Appl. 122 (2002), 157-181. https://doi.org/10.1016/S0166-8641(01)00141-9
V. Gutev, Fell continuous selections and topologically well-orderable spaces II, Proceedings of the Ninth Prague Topological Symposium (2001), Topology Atlas, Toronto, 2002, pp. 157-163 (electronic).
V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan 58 (2006), no. 4, 1203-1210. https://doi.org/10.2969/jmsj/1179759545
V. Gutev, Weak orderability of second countable spaces, Fund. Math. 196 (2007), no. 3, 275-287. https://doi.org/10.4064/fm196-3-4
V. Gutev, Selections and hyperspaces, Recent progress in general topology III (K. P. Hart, J. van Mill, and P. Simon, eds.), Atlantis Press, Springer, 2014, pp. 535-579. https://doi.org/10.2991/978-94-6239-024-9_12
V. Gutev, Selections and approaching points in products, Comment. Math. Univ. Carolin. 57 (2016), no. 1, 89-95. https://doi.org/10.14712/1213-7243.2015.147
V. Gutev and T. Nogura, Selections and order-like relations, Appl. Gen. Topol. 2 (2001), 205-218. https://doi.org/10.4995/agt.2001.2150
V. Gutev and T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001), 2809-2815. https://doi.org/10.1090/S0002-9939-01-05883-X
V. Gutev and T. Nogura, Fell continuous selections and topologically well-orderable spaces, Mathematika 51 (2004), 163-169. https://doi.org/10.1112/S002557930001559X
V. Gutev and T. Nogura, Selection pointwise-maximal spaces, Topology Appl. 146-147 (2005), 397-408. https://doi.org/10.1016/j.topol.2003.06.004
V. Gutev and T. Nogura, Weak orderability of topological spaces, Topology Appl. 157 (2010), 1249-1274. https://doi.org/10.1016/j.topol.2009.07.015
H. Kok, Connected orderable spaces, Mathematisch Centrum, Amsterdam, 1973, Mathematical Centre Tracts,no. 49.
E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. https://doi.org/10.1090/S0002-9947-1951-0042109-4
E. Michael, Cuts, Acta Math. 111 (1964), 1-36. https://doi.org/10.1007/BF02391006
E. Michael, J-spaces, Topology Appl. 102 (2000), no. 3, 315-339. https://doi.org/10.1016/S0166-8641(99)00237-0
J. C. Oxtoby, Cartesian products of Baire spaces, Fund. Math. 49 (1960), 157-166. https://doi.org/10.4064/fm-49-2-157-166
B. E. Šapirovskiĭ, The imbedding of extremally disconnected spaces in bicompacta. b-points and weight of pointwise normal spaces, Dokl. Akad. Nauk SSSR 223 (1975), no. 5, 1083-1086 (in Russian).
