Connectedness in monotone spaces.
In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of...
To overcome the somewhat artificial difficulties in classical optimization theory concerning the existence and stability of minimizers, a new setting of constrained optimization problems (called problems with tolerance) is proposed using given proximity structures to define the neighbourhoods of sets. The infimum and the so-called minimizing filter are then defined by means of level sets created by these neighbourhoods, which also reflects the engineering approach to constrained optimization problems....
We study topological properties of Valdivia compact spaces. We prove in particular that a compact Hausdorff space K is Corson provided each continuous image of K is a Valdivia compactum. This answers a question of M. Valdivia (1997). We also prove that the class of Valdivia compacta is stable with respect to arbitrary products and we give a generalization of the fact that Corson compacta are angelic.