Fully-closed mapping

A continuous mapping with the following property: For any point and for any finite family of open subsets of the space such that , the set is open. Here denotes the small image of the set under the mapping . Any fully-closed mapping is closed. The inequality is valid for any fully-closed mapping of a normal space . For this reason, fully-closed mappings can be employed to isolate fairly wide classes of compacta with non-coinciding dimensions dim and ind. Moreover, irrespective of the multiplicity of the mapping . Let , let be a fully-closed mapping and let be the decomposition of the elements of which are all pre-images of the points, and all points of . Then, for a regular space , the quotient space of with respect to the decomposition is also regular; this property is characteristic of fully-closed mappings in the class of closed mappings.