Difference between revisions of "Pseudo-open mapping"
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| + | A [[Continuous mapping|continuous mapping]] $ f : X \rightarrow Y $ | ||
| + | such that for every point $ y \in Y $ | ||
| + | and any neighbourhood $ U $ | ||
| + | of the set $ f ^ { - 1 } y $ | ||
| + | in $ X $ | ||
| + | it is always true that $ y \in \mathop{\rm Int} f U $( | ||
| + | here $ \mathop{\rm Int} f U $ | ||
| + | is the set of all interior points of $ f U $ | ||
| + | with respect to $ Y $). | ||
====Comments==== | ====Comments==== | ||
| − | It is also called a hereditarily quotient mapping, because a mapping | + | It is also called a hereditarily quotient mapping, because a mapping $ f: X \rightarrow Y $ |
| + | is pseudo-open if and only if for every $ B \subseteq Y $ | ||
| + | the corestriction $ f _ {B} : f ^ { - 1 } [ B] \rightarrow B $ | ||
| + | is a [[Quotient mapping|quotient mapping]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR></table> | ||
Latest revision as of 08:08, 6 June 2020
A continuous mapping $ f : X \rightarrow Y $
such that for every point $ y \in Y $
and any neighbourhood $ U $
of the set $ f ^ { - 1 } y $
in $ X $
it is always true that $ y \in \mathop{\rm Int} f U $(
here $ \mathop{\rm Int} f U $
is the set of all interior points of $ f U $
with respect to $ Y $).
Comments
It is also called a hereditarily quotient mapping, because a mapping $ f: X \rightarrow Y $ is pseudo-open if and only if for every $ B \subseteq Y $ the corestriction $ f _ {B} : f ^ { - 1 } [ B] \rightarrow B $ is a quotient mapping.
References
| [a1] | R. Engelking, "General topology" , Heldermann (1989) |
How to Cite This Entry:
Pseudo-open mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-open_mapping&oldid=48349
Pseudo-open mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-open_mapping&oldid=48349
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article