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Difference between revisions of "Pseudo-open mapping"

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A [[Continuous mapping|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757601.png" /> such that for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757602.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757603.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757604.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757605.png" /> it is always true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757606.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757607.png" /> is the set of all interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757608.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p0757609.png" />).
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A [[Continuous mapping|continuous mapping]]  $  f :  X \rightarrow Y $
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such that for every point  $  y \in Y $
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and any neighbourhood  $  U $
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of the set  $  f ^ { - 1 } y $
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in  $  X $
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it is always true that  $  y \in  \mathop{\rm Int}  f U $(
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here  $  \mathop{\rm Int}  f U $
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is the set of all interior points of  $  f U $
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with respect to  $  Y  $).
  
 
====Comments====
 
====Comments====
It is also called a hereditarily quotient mapping, because a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p07576010.png" /> is pseudo-open if and only if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p07576011.png" /> the corestriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075760/p07576012.png" /> is a [[Quotient mapping|quotient mapping]].
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It is also called a hereditarily quotient mapping, because a mapping $  f: X \rightarrow Y $
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is pseudo-open if and only if for every $  B \subseteq Y $
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the corestriction $  f _ {B} : f ^ { - 1 } [ B] \rightarrow B $
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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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article