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It is known that is always zero-dimensional and completely regular. Roy in, is the set of all nonempty finite subsets of with the topology generated by the sets of the form, where, and is an open subset in containing. The Pixley–Roy hyperspace over a space, defined by C. Moreover, if is a family of subsets of a space and, then we set. denotes the set of all positive integers, and the first infinite ordinal is denoted by. Throughout this paper, all spaces are assumed to be at least. By the abovementioned results, we show that is second-countable but is not a -space for each. However, he does not know whether or not is a -space for each. Sakai proved that for each, is sequential but is not Fréchet–Urysohn. Moreover, we show that there exists a space with a countable -network, but and with do not have point-countable -networks. We also obtain that if is regular and is a cosmic space (resp., -space, strict -space, and stric -space), then is also a cosmic space (resp., -space, strict -space, stric -space). Lin in 2019 on Pixley–Roy hyperspaces and obtained the following results: (1) If has a point-countable -network (resp., strict Pytkeev network), then so does (2) If has a countable -network (resp., strict Pytkeev network), then so does Banakh in 2015, and the notion of -network that was introduced by X. Next, we study the notion of strict Pytkeev network which was introduced by T. Furthermore, if has a countable -network, then also has a countable -network.
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If is regular with a point-countable -network, then so does. We prove that if has a point-countable -network, then also has a point-countable -network. In this paper, we study relationships between some generalized metric properties of a space and generalized metric properties of Pixley–Roy hyperspaces over. Especially, they studied the relation between a space satisfying such a property and its Pixley-Roy hyperspaces satisfying the same property. The topological and generalized metric properties on Pixley–Roy hyperspaces have been studied by many authors from different points of view, as shown in. Furthermore, the Pixley-Roy hyperspace is not a -space for each. By these results, we obtain that if the Pixley–Roy hyperspace is a cosmic space (resp., -space, strict -space, and stric -space), then so is.
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On the other hand, we show that if the Pixley–Roy hyperspace has a countable -network (resp., -network and strict Pytkeev network), then so does. Moreover, if has a point-countable -network (resp., strict Pytkeev network), then the Pixley–Roy hyperspace also has a point-countable -network (resp., strict Pytkeev network). If is a regular space with a point-countable -network, then so does the Pixley-Roy hyperspace.
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Math., 47(1973), 553-565.In this paper, we prove that if a space has a point-countable -network, then the Pixley-Roy hyperspace also has a point-countable -network. Schirmer, On fixed point sets of homeomorphisms of the n-ball, Israel J. Schirmer, Fixed point sets of homeomorphisms of compact surfaces, Israel J. Schirmer, Fixed point sets of continuous selfmaps, in: Fixed Point Theory, Conf. 49, Marcel Dekker, New York and Basel, 1978. Nadler Jr., Hyperspaces of sets: A text with research questions, Monographs and Textbooks, Pure Appl. Weiss, Fixed point sets of metric and nonmetric spaces, Trans. Tymchatyn, Fixed point sets of products and cones, Pacific J. Nadler Jr., Examples and questions in the theory of fixed point sets, Canad. Martin, Fixed point sets of homeomorphisms of metric products, Proc. Curtis, Hyperspaces of noncompact metric spaces, Compositio Mathematica, 40(1980), 126-130. Martin, Compact groups and fixed point sets, Trans. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, London, 1972. Voytsitskyy, Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts, Topology Appl., 154(2007), 2009-2025. Srivastava, On S-equivariant complete invariance property, Journal of the Indian Math. Antonyan, West's problem on equivariant hyperspaces and Banach-Mazur compacta, Trans.