In the paper the singular Cauchy-Nicoletti problem for the system ot two ordinary differential equations is considered. New sufficient conditions for solvability of this problem are proved. In the proofs the topological method is applied. Some comparisons with known results are also given in the paper.
The relativization of Gryzlov’s theorem about the size of compact -spaces with countable pseudocharacter is false.
A homeomorphism h:X → X of a compactum X is expansive provided that for some fixed c > 0 and any distinct x, y ∈ X there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a circle-like continuum that admits an expansive homeomorphism, then X is homeomorphic to a solenoid.
The hyperspaces and in consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute -spaces and that, indeed, they are not -spaces. The main result is that is an absorber for the class of all absolute -spaces and is therefore homeomorphic to the standard model space of this class.
Let be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that for every n > 1 whereas .