Topologies of closed subsets

Weak Topologies for the Closed Subsets of a Metrizable Space

The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies—including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence—arise as weak topologies generated by ...

Topologies on Spaces of Subsets

SOME REMARKS ON GENERALIZATIONS OF MULTIPLICATIVELY CLOSED SUBSETS

Let R be a commutative ring with identity and Mbe a unitary R-module. In this paper we generalize the conceptmultiplicatively closed subset of R and we study some propertiesof these genaralized subsets of M. Among the many results in thispaper, we generalize some well-known theorems about multiplicativelyclosed subsets of R to these generalized subsets of M. Alsowe show that some other well-kno...

Topologies Generated by Closed Intervals

If 〈L, <〉 is a dense linear ordering without end points and A and B disjoint dense subsets of L, then the topology OAB on the set L generated by closed intervals [a, b], where a ∈ A and b ∈ B, is finer than the standard topology, O<, generated by all open intervals and 〈L,OAB〉 is a GO-space. The basic properties of the topology OAB (separation axioms, cardinal functions, metrizability) are inve...

Topologies on IR” induced by smooth subsets

Ancel, F.D., Topologies on Iw” induced by smooth subsets, Topology and its Applications 43 (1992) 189-201. If Y is a collection of subsets of iw”, let Y!? denote the largest topology on Iw” which restricts to the standard topology on each element of Y, and let ?t$ denote the homeomorphism group of Iw” wtth the topology &. Let Y_, denote the standard topology on Iw” and let %?_d denote the homeo...