We give a short proof of the theorem that any family of subsets of R with the property that the intersection of any non empty nite subfamily can be represented as the disjoint union of at most k closed convex sets has Helly number at most k d

In this paper, we present a new type of set called $\Psi_{\Gamma}-C$ by using the operator $\Psi_{\Gamma}$. We investigate relationships these sets with some special which were studied in literature. For instance $\theta$-open set, semi $\theta$-semiopen regular $\theta$-closed set. particular, show that is weaker than Furthermore, prove collection closed under arbitrary union. Finally, obtain ...

Let ω(·) denote the union of all ω-limit sets of a given map. As the main result of this paper we prove that, for given continuous interval maps f1, . . . , fm, the union of all ω-limit sets of the product map f1 × · · · × fm and the cartesian product of the sets ω(f1), . . . , ω(fm) coincide. This result enriches the theory of multidimensional permutation product maps, i.e., maps of the form F...

A set X IR d is n-convex if among any n its points there exist two such that the segment connecting them is contained in X. Perles and Shelah have shown that any closed (n + 1)-convex set in the plane is the union of at most n 6 convex sets. We improve their bound to 18n 3 , and show a lower bound of order (n 2). We also show that if X IR 2 is an n-convex set such that its complement has one-po...

‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...