Аннотация:Every boundedly compact m-connected (Menger-connected) set is shown to be monotone path-connected and to be a sun in a broad class of Banach spaces (in particular, in separable spaces). The intersection of a boundedly compact monotone path-connected (m-connected) set with a closed ball is shown to be cell-like (of trivial shape) and, in particular, acyclic (contractible in the finite-dimensional case) and is a sun. It also prove that every boundedly weakly compact m-connected set is monotone path-connected. The Rainwater–Simons weak convergence theorem is extended to the case of convergence with respect to the Brown-associated norm.