View source for Fully-closed mapping

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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|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.