The relationships between topologies and generalized rough sets Abstract   Topology can be used as a tool for knowledge discovery in databases and has been studied intensively in rough set theory. Topology and binary relations can be induced from each other. In this paper, we mainly study topologies induced by arbitrary binary relations. We first consider the elementary properties of such topologies, show that the relation induced by the topology      xcfx84    R      from a relation R is exactly the transitive closure of    I  xe2x88xaa  R   , where I is the relation of equality, and that the topology induced by the relation      R    xcfx84      from a topology xcfx84 is exactly xcfx84 if and only if xcfx84 is quasidiscrete. Then we examine special topological properties of      xcfx84    R      induced by a binary relation R. In particular, we obtain equivalent descriptions of separation axioms and provide characterizations of topological properties such as compactness, second countability and connectedness.

Dismantlability, connectedness, and mixing in relational structures. The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involvedlike, for example, dismantlabilityand their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, in approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions and free energy.  :125,spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceno. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FOdefinability, studied by Larose, Loten, and Tardif.