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On the n -fold symmetric product of a space with a σ - ( P ) -property c n -network ( c k -network)

Luong Q. TuyenOng V. Tuyen — 2020

Commentationes Mathematicae Universitatis Carolinae

We study the relation between a space X satisfying certain generalized metric properties and its n -fold symmetric product n ( X ) satisfying the same properties. We prove that X has a σ - ( P ) -property c n -network if and only if so does n ( X ) . Moreover, if X is regular then X has a σ - ( P ) -property c k -network if and only if so does n ( X ) . By these results, we obtain that X is strict σ -space (strict -space) if and only if so is n ( X ) .

About w c s -covers and w c s * -networks on the Vietoris hyperspace ( X )

Luong Quoc TuyenOng V. TuyenPhan D. TuanNguzen X. Truc — 2023

Commentationes Mathematicae Universitatis Carolinae

We study some generalized metric properties on the hyperspace ( X ) of finite subsets of a space X endowed with the Vietoris topology. We prove that X has a point-star network consisting of (countable) w c s -covers if and only if so does ( X ) . Moreover, X has a sequence of w c s -covers with property ( P ) which is a point-star network if and only if so does ( X ) , where ( P ) is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other...

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