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.
Fully-closed mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fully-closed_mapping&oldid=47007