The hyperspace of k-dimensional closed convex sets

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-02-01 DOI:10.1016/j.topol.2024.109154

Adriana Escobedo-Bustamante , Natalia Jonard-Pérez

引用次数: 0

Abstract

For every

n \geq 2

, let

K_{k}^{n}

denote the hyperspace of all k-dimensional closed convex subsets of the Euclidean space

R^{n}

endowed with the Atouch-Wets topology. Let

K_{k, b}^{n}

be the subset of

K_{k}^{n}

consisting of all k-dimensional compact convex subsets. In this paper we explore the topology of

K_{k}^{n}

and

K_{k, b}^{n}

and the relation of these hyperspaces with the Grassmann manifold

G_{k} (n)

. We prove that both

K_{k}^{n}

and

K_{k, b}^{n}

are Hilbert cube manifolds with a fiber bundle structure over

G_{k} (n)

. We also show that the fiber of

K_{k, b}^{n}

with respect to this fiber bundle structure is homeomorphic with

R^{\frac{k (k + 1) + 2 n}{2}} \times Q

, where Q stands for the Hilbert cube.

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.