Irreducible mapping
A continuous mapping of a topological space onto a topological space
such that the image of every closed set in
, other than
itself, is different from
. If
is a continuous mapping,
, and if all inverse images of points under
are compact, then there exists a closed subspace
in
such that
and such that the restriction of
to
is an irreducible mapping. The combination of the requirements on a mapping of being irreducible and being closed has an outstanding effect: Spaces linked by such mappings do not differ in a number of important characteristics; in particular, they have the same Suslin number and
-weight. But the main value of closed irreducible mappings lies in the central role they play in the theory of absolutes.
References
[1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |
Comments
See Absolute.
References
[a1] | J.R. Porter, R.G. Woods, "Extensions and absolutes of Hausdorff spaces" , Springer (1988) |
Irreducible mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_mapping&oldid=42076