Fully-closed mapping

A continuous mapping $ f: X \rightarrow Y $ with the following property: For any point $ y \in Y $ and for any finite family $ \{ O _ {1}, \dots, O _ {s} \} $ of open subsets of the space $ X $ such that $ f ^ { - 1 } ( y) = {\cup _ {i=1} ^ {s} } O _ {i} $, the set $ \{ y \} \cup ( {\cup _ {i=1} ^ {s} } f ^ { \sharp } O _ {i} ) $ is open. Here $ f ^ { \sharp } O _ {i} $ denotes the small image of the set $ O _ {i} $ under the mapping $ f $. Any fully-closed mapping is closed. The inequality $ \mathop{\rm dim} X \leq \max \{ \mathop{\rm dim} Y , \mathop{\rm dim} f \} $ is valid for any fully-closed mapping $ f: X \rightarrow Y $ of a normal space $ X $. For this reason, fully-closed mappings can be employed to isolate fairly wide classes of compacta with non-coinciding dimensions dim and ind. Moreover, $ \mathop{\rm dim} Y \leq \mathop{\rm dim} X+ 1 $ irrespective of the multiplicity of the mapping $ f $. Let $ y \in Y $, let $ f: X \rightarrow Y $ be a fully-closed mapping and let $ R( f, y) $ be the decomposition of $ X $ the elements of which are all pre-images $ f ^ { - 1 } ( y ^ \prime ) $ of the points, and all points of $ f ^ { - 1 } ( y) $. Then, for a regular space $ X $, the quotient space $ X _ {f} ^ {y} $ of $ X $ with respect to the decomposition $ R( f, y) $ is also regular; this property is characteristic of fully-closed mappings in the class of closed mappings.