The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology
The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology
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DOI：
10.4064/ba55-2-4
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发表时间：
2007
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影响因子：
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通讯作者：
K. Sakai;Zhongqiang Yang
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文献类型：
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作者：
K. Sakai;Zhongqiang Yang
令ConvF (R) 为具有Fell 拓扑的欧几里得空间R 中所有非空闭凸集的空间。在本文中，我们证明对于每个 n > 1，ConvF (R) ≈ R × Q，而 ConvF (R) ≈ R× I。令 Conv(X) 为赋范线性空间 X = (X, ‖·‖) 中所有非空闭凸集的集合。我们可以考虑 Conv(X) 上的各种拓扑。在论文[6]中，研究了具有Hausdorff度量拓扑、Attouch-Wets拓扑和Wijsman拓扑的空间Conv(X)的AR性质。在本文中，我们将考虑 Conv(X) 上的 Fell 拓扑，它是由 U− = {A ∈ Conv(X) | 形式的集合生成的。 A ∩ U 6= ∅} 且 (X \K)+ = {A ∈ Conv(X) | A ⊂ X \K}，其中 U 在 X 中是开集，K 是紧集。该拓扑也在集合 Conv*(X) = Conv(X) ∪ {∅} 上定义。通过 ConvF (X) 和 ConvF (X)，我们表示承认 Fell 拓扑的空间 Conv*(X) 和 Conv(X)。如果 X 是有限维（等效于局部紧致），则 ConvF (X) 是局部紧致可度量空间，ConvF (X) 是其 Alexandrorff 单点紧致化。容易看出 ConvF ((0, 1)) 同胚于 (≈) 去除两个顶点的三角形 Δ \ {(0, 0), (1, 1)}，其中 Δ = {(x, y) ∈ I | x 6 y} ⊂ I。由于 ConvF (R) ≈ ConvF ((0, 1))，我们有 ConvF (R) ≈ Δ \ {(0, 0), (1, 1)} ≈ R× I，因此 ConvF (R) ≈ Δ/{(0, 0), (1, 1)} ≈ (S × I)/({pt} × I)， 1991年 数学学科分类。 54B20、54D05、54E45、57N20。
Let ConvF (R) be the space of all non-empty closed convex sets in Euclidean space R endowed with the Fell topology. In this paper, we prove that ConvF (R) ≈ R × Q for every n > 1 whereas ConvF (R) ≈ R× I. Let Conv(X) be the set of all non-empty closed convex sets in a normed linear space X = (X, ‖·‖). We can consider various topologies on Conv(X). In the paper [6], the AR-property of the spaces Conv(X) with the Hausdorff metric topology, the Attouch-Wets topology, and the Wijsman topology has been studied. In this paper, we shall consider the Fell topology on Conv(X), which is generated by the sets of the form U− = {A ∈ Conv(X) | A ∩ U 6= ∅} and (X \K)+ = {A ∈ Conv(X) | A ⊂ X \K}, where U is open and K is compact in X. This topology is also defined on the set Conv∗(X) = Conv(X) ∪ {∅}. By ConvF (X) and ConvF (X), we denote the spaces Conv∗(X) and Conv(X) admitting the Fell topology. In case X is finite-dimensional (equivalently locally compact), ConvF (X) is a locally compact metrizable space and ConvF (X) is its Alexandorff onepoint compactification. It is easy to see that ConvF ((0, 1)) is homeomorphic to (≈) the triangle with two vertices removed, ∆ \ {(0, 0), (1, 1)}, where ∆ = {(x, y) ∈ I | x 6 y} ⊂ I. Since ConvF (R) ≈ ConvF ((0, 1)), we have ConvF (R) ≈ ∆ \ {(0, 0), (1, 1)} ≈ R× I, hence it follows that ConvF (R) ≈ ∆/{(0, 0), (1, 1)} ≈ (S × I)/({pt} × I), 1991 Mathematics Subject Classification. 54B20, 54D05, 54E45, 57N20.