Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bangbang property is proved for a new class of “concave” multifunctions, characterized by the existence of suitable linear selections. The proofs rely ...

In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid End(A) of a countable relational structure A. As applications, show that monoids several well-known structures, including random graph, directed partial order, possess unique Polish topology. every case topology is subspace induced by usual Baire space NN. We also m...

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a “good” additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are consid...

Let k be the least cardinal such that the real Une can be covered by k many nowhere dense sets. We show that k can be characterized as the least cardinal such that "infinitely equal" reals fail to exist for families of cardinality k. Let Baire (k) stand for: "The real line is not the union of k many nowhere dense sets (a set is nowhere dense iff its closure has no interior)". The property was e...

A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that orbit induces generically an embedding the associated Becker graphs. introduce notion dimension for Polish $G$-spaces preserved under $*$-reductions. For every natural number $n$ we define free action $S_{\infty}$ whose on inv...

Calude, C., Relativized topological size of sets of partial recursive functions (Note), Theoretical Computer Science 87 (1991) 347-352. In [ 11, a recursive topology on the set of unary partial recursive functions was introduced and recursive variants of Baire topological notions of nowhere dense and meagre sets were defined. These tools were used to measure the size of some classes of partial ...

The concept of a bipolar metric space is presented in this article. In fuzzy space, the notions continuous t-norms and symmetry $$t_s$$ -conorms employs important role. Requirements triangular norms are used to extrapolate with probability density function triangle inequality. Dual processes known as conorms. Continuous define paper. Besides, number topological functional properties have been s...

Using the notion of formal ball, we present a few new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its d-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower sem...