The manifold of topologies and the topology of a manifold (Q1917801)

The author considers the dynamical polysystems (DP), i.e., the families of \(C^\infty\)-smooth vector fields on a smooth manifold, and the topologies which are generated by them on the set of points of that manifold. The author states that the investigation of ``smooth'' objects, in particular of DP, leads to problems which should be considered within the framework of the set-theoretic topology. Such are, for example, the problems of the structure of reachable sets and that of their boundaries, the problems of limit sets of trajectories, of generalized foliations etc. On the other hand (the author states that) some interesting examples of the set-theoretic topology can be constructed by DP. For example one can prove that, for any integer \(n\geq 2\), there exist an ``almost Euclidean'' topology \(\tau\) of \(\mathbb{R}^n\) such that the topological space \((\mathbb{R}^n, \tau)\) is a connected and two-dimensional metric space.