No abstract

<p>Let T = (<strong>T</strong>, ≤) and T₁= (<strong>T</strong>₁ , ≤₁) be linearly ordered sets and X be a topological space.  The main result of the paper is the following: If function ƒ(t,x) : <strong>T</strong> × X → <strong>T</strong>₁is continuous in each  variable (“t” and  “x”)  separately  and  function ƒ_x(t)  = ƒ(t,x) is  monotonous  on <strong>T</strong> for  every x ∈ X,  then ƒ is  continuous  mapping  from<strong> T</strong> × X to <strong>T</strong>₁,  where <strong>T</strong> and <strong>T</strong>₁ are  considered  as  topological  spaces  under  the order topology and <strong>T</strong> × X is considered as topological space under the Tychonoff topology on the Cartesian  product of topological spaces <strong>T</strong> and X.</p>

The notion of oriented set is the basic elementary concept of the theory of changeable sets. The main motivation for the introduction of changeable sets was the sixth Hilbert problem, that is, the problem of mathematically rigorous formulation of the fundamentals of theoretical physics. In the present paper the necessary and sufficient condition of the existence of one-point time on an oriented set is established. From the intuitive point of view, one-point time is the time associated with the evolution of a system consisting of only one object (for example, from one material point). Namely, it is proven that the one-point time exists on the oriented set if and only if this oriented set is a quasi-chain. Also, using the obtained result, the problem of describing all possible images of linearly ordered sets is solved. This problem naturally arises in the theory of ordered sets.

The original Hassani transforms were introduced in the works of Algerian physicist M.E. Hassani. Hassani's generalized (superluminal) kinematics appeared in the cohntext of generalization and development of Hassani's ideas. In the present paper, applying Theorem of non returning for universal kinematics, it is proven that Hassani's generalized kinematics with positive direction of time are certainly time irreversible. From the physical point of view the last result means that in any time-positive Hassani kinematics temporal paradoxes are impossible basically, that is there is not potential possibility to affect the own past by means of "traveling" and "jumping" between reference frames.