Difference between revisions of "Irreducible mapping"
From Encyclopedia of Mathematics
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| − | A [[ | + | A [[continuous mapping]] of a topological space $X$ onto a topological space $Y$ such that the image of every closed set in $X$, other than $X$ itself, is different from $Y$. If $f : X \rightarrow Y$ is a continuous mapping, $f(X) = Y$, and if all inverse images of points under $f$ are compact, then there exists a closed subspace $X_1$ in $X$ such that $f(X_1) = Y$ and such that the restriction of $f$ to $X_1$ 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 [[cardinal characteristic]]s; in particular, they have the same [[Suslin number]] and $\pi$-weight. But the main value of closed irreducible mappings lies in the central role they play in the theory of absolutes. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
====Comments==== | ====Comments==== | ||
| − | See [[ | + | See [[Absolute]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.R. Porter, R.G. Woods, "Extensions and absolutes of Hausdorff spaces" , Springer (1988)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.R. Porter, R.G. Woods, "Extensions and absolutes of Hausdorff spaces" , Springer (1988)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 19:22, 14 October 2017
A continuous mapping of a topological space $X$ onto a topological space $Y$ such that the image of every closed set in $X$, other than $X$ itself, is different from $Y$. If $f : X \rightarrow Y$ is a continuous mapping, $f(X) = Y$, and if all inverse images of points under $f$ are compact, then there exists a closed subspace $X_1$ in $X$ such that $f(X_1) = Y$ and such that the restriction of $f$ to $X_1$ 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 cardinal characteristics; in particular, they have the same Suslin number and $\pi$-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) |
How to Cite This Entry:
Irreducible mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_mapping&oldid=16891
Irreducible mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_mapping&oldid=16891
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article