When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies
When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies
复制标题
DOI：
10.1016/0166-8641(95)00098-4
复制
发表时间：
1996-06
影响因子：
0.6
通讯作者：
T. Nogura;D. Shakhmatov
中科院分区：
文献类型：
--
作者：
T. Nogura;D. Shakhmatov
对于给定的拓扑空间X，我们考虑X的所有闭子集的超空间F（X）上的两个拓扑。F（X）上的Fell拓扑TF是由族{OVK：V在X中是开的且K <$X是紧的}作为子基生成的，其中OVK= {F <$F（X）：F <$V <$$>且F <$K =<$}。拓扑TF总是紧的，与空间X无关。Kuratowski拓扑TK是F（X）上的最小拓扑，其包含由族{{F}生成的下Vietoris拓扑TIV：Φ <$F（X）}作为子基，以及上Kuratowski拓扑TuK，这是F（X）上的最强拓扑，使得X的闭子集的任意网的上KuratowskiPainlevé收敛到某个闭集A意味着同一个网，被认为是拓扑空间（F（X），TuK）的点的网，在这个空间中收敛到点A。[回想一下，如果<${<${Aμ：μ <$λ}：λ <$Λ}<$A，则净<$Aλ <$λ <$<$F（X）上Kuratowski-Painlevé收敛于A。]包含TF = TK对任意空间X成立，而方程TF= TK等价于X的调和，这是Dolecki，Greco和Lechicki最近引入的概念。这三位作者证明了完备度量空间是和谐的。本文给出了具有Baire性质的度量空间的一个例子。我们还证明了和谐是一个微妙的属性，通过提供一个例子，两个辅音空间X和Y，使他们的不相交的工会X Y是不和谐的。特别地，局部辅音空格不需要是辅音的。
For a given topological space X we consider two topologies on the hyperspace F(X) of all closed subsets of X. The Fell topology TFon F(X) is generated by the family {OVK: V is open in X and K ⊆ X is compact} as a subbase, where OVK= {FϵF(X): F ∩ V ≠ Ø and F ∩ K = Ø}. The topology TFis always compact, regardless of the space X. The Kuratowski topology TKis the smallest topology on F(X) which contains both the lower Vietoris topology TlV, generated by the family {{FϵF(X): F \ Φ ≠ Ø}: ΦϵF(X)} as a subbase, and the upper Kuratowski topology TuK, which is the strongest topology on F(X) such that upper KuratowskiPainlevé convergence of an arbitrary net of closed subsets of X to some closed set A implies that the same net, considered as a net of points of the topological space (F(X), TuK) , converges in this space to the point A. [Recall that a net 〈Aλ〉λϵΛ⊆ F(X) upper Kuratowski-Painlevé converges to A if ∩{ ∪{Aμ: μ ⩾ λ} : λ ϵ Λ} ⊆ A .] The inclusion TF⊆ TKholds for an arbitrary space X, while the equation TF= TKis equivalent to consonance of X, the notion recently introduced by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Baire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant spaces X and Y such that their disjoint union X ⊕ Y is not consonant. In particular, locally consonant spaces need not be consonant.