On the Space of the Solutions of an Ordinary Differential Equation

In this article, are considered the ordinary differential equations with continuous right-hand sides. Some topological, cardinal, and homotopy properties of the space of solutions of ordinary differential equations are studied. Our work includes compactness, density, weight, network weight, Souslin number, tightness, path-connectedness, contractibility, and some functorial properties of the space of solutions of ordinary differential equations with continuous right-hand sides. We prove that the space of solutions of ordinary differential equations is compact, if and only if every sequence of solutions admits a uniformly convergent subsequence on compact intervals. Also shown that the space of solutions of ordinary differential equations with continuous function is separable and the solution has countable tightness. Besides, proved that the weight, network weight, and Souslin number of the space of solutions of ordinary differential equations are continuum. Moreover, it is proved that the space of solutions of ordinary differential equations with continuous right-hand side is path-connected and contractible. Finally, studied some functorial properties of the space of solutions of ordinary differential equations with continuous right-hand side under the functors idempotent probability measures and symmetric power.