# 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.

**How to Cite This Entry:**

Fully-closed mapping.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fully-closed_mapping&oldid=51640