# 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 $\{ y \} \cup ( {\cup _ {i=} 1 ^ {s} } f ^ { \# } O _ {i} )$ is open. Here $f ^ { \# } O _ {i}$ 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.