Geodesic
Hyperspace of the $\Pi$-complete spaces and maps
Eurasian mathematical journal, Tome 12 (2021) no. 2, pp. 104-110 
A. A. Zaitov; D. I. Jumaev. Hyperspace of the $\Pi$-complete spaces and maps. Eurasian mathematical journal, Tome 12 (2021) no. 2, pp. 104-110. http://geodesic.mathdoc.fr/item/EMJ_2021_12_2_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper we establish that the space $\exp_\beta X$ of compact subsets of a Tychonoff space $X$ is $\Pi$-complete if and only if $X$ is so. Further, we prove the Tychonoff map $\exp_\beta f: \exp_\beta X\to \exp_\beta Y$ is $\Pi$-complete if and only if a given map $f: X\to Y$ is $\Pi$-complete.

[1] A. V. Arkhangelsky, V. I. Ponomarev, Fundamentals of the general topology: problems and exercises, D. Reidel Publishing Company, 1983 ; Originally published as: A. V. Arkhangelsky, V. I. Ponomarev, Osnovy Obsheii Topologii v Zadachakh i Uprajneniyakh, Nauka, M., 1974 | MR

[2] A. A. Borubaev, A. A. Chekeev, “$\tau$-Completeness of topological groups”, Journal of Mathematical Sciences, 81:2 (1996), 2517–2523 | DOI | MR

[3] V. V. Fedorchuk, V. V. Filippov, General topology. Basic constructions, Fizmatlit, M., 2006 (in Russian)

[4] D. K. Musayev, B. A. Pasynkov, On compactness and completeness properties of topological spaces and continuous maps, Fan, Tashkent, 1994 (in Russian)

[5] D. K. Musayev, On compactness and completeness properties of topological spaces and continuous maps, NisoPoligraf, Tashkent, 2011 (in Russian)

[6] Yu. V. Sadovnichii, “Lifting the functors $U_\tau$ and $U_R$ to the categories of bounded metric spaces and uniform spaces”, Sb. Math., 191:11 (2000), 1667–1691 | DOI | MR | Zbl

[7] A. A. Zaitov, D. I. Jumaev, “Hyperspaces of superparacompact spaces and continuous maps”, Universal Journal of Mathematics and Applications, 2:2 (2019), 8 pp., arXiv: 1811.05347