On convergence of mappings in metric spaces with direct and inverse modulus conditions

For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of the uniform limit of these mappings. It is proved that, the uniform limit of these mappings is light mapping, whenever a function which corresponds to distortion of families of curves, is of finite mean oscillation at every point. Besides that, for one class of homeomorphisms of metric spaces, there are obtained theorems about equicontinuity of inverse mappings.