Geodesic
Categorical and Cardinal Properties of Hyperspaces with a Finite Number of Components
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Problems of Modern Topology and its Applications», Tome 144 (2018), pp. 96-103

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In this paper, we examine categorical and cardinal properties of hyperspaces with finite number of components. We prove that the functor $C_{n}:\operatorname{Comp} \to \operatorname{Comp}$ is not normal, i.e., it does not preserve epimorphisms of continuous mappings. We also discuss the density, the caliber, and the Shanin number of the space $C_{n}(X)$.
Keywords: category, functor, hyperspace, connected component, density, caliber. 
@article{INTO_2018_144_a10,
     author = {R. B. Beshimov and N. K. Mamadaliev and Sh. Kh. Eshtemirova},
     title = {Categorical and {Cardinal} {Properties} of {Hyperspaces} with a {Finite} {Number} of {Components}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {96--103},
     publisher = {mathdoc},
     volume = {144},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_144_a10/}
}
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R. B. Beshimov; N. K. Mamadaliev; Sh. Kh. Eshtemirova. Categorical and Cardinal Properties of Hyperspaces with a Finite Number of Components. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Problems of Modern Topology and its Applications», Tome 144 (2018), pp. 96-103. http://geodesic.mathdoc.fr/item/INTO_2018_144_a10/