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''of a space''
 
''of a space''
  
A countable family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383201.png" /> of coverings of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383202.png" /> by open sets in an ambient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383203.png" /> such that
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A countable family $P$ of coverings of a space $X$ by open sets in an ambient space $Y$ such that
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$$
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\bigcap \{ \mathrm{St}_\gamma(x) : \gamma \in P \} \subset X
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$$
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for every point $x \in X$; here $\mathrm{St}_\gamma(x)$ denotes the star of the point $x$ relative to $\gamma$, i.e. the union of all elements of $\gamma$ containing the point $x$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383204.png" /></td> </tr></table>
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The concept of a feathering forms the basis of the definition of the so-called $P$-space (in the sense of A.V. Arkhangel'skii). A space $X$ is called a $P$-space if it has a feathering in its [[Stone–Čech compactification]] or [[Wallman compactification]]. Every complete space (in the sense of Čech) is a $P$-space. Every $P$-space has pointwise countable type. In a $P$-space, the [[addition theorem]] for [[Weight of a topological space|weight]] holds and the net weight coincides with the weight. Paracompact $P$-spaces are perfect pre-images of metric spaces. Paracompact $P$-spaces with a pointwise countable base are metrizable, just as spaces of this type with a $G_\delta$-diagonal are also metrizable. The perfect image and the perfect pre-image of a paracompact $P$-space are also paracompact $P$-spaces.
 
 
for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383205.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383206.png" /> denotes the star of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383207.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383208.png" />, i.e. the union of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f0383209.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832010.png" />).
 
 
 
The concept of a feathering forms the basis of the definition of the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832011.png" />-space (in the sense of A.V. Arkhangel'skii). A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832012.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832014.png" />-space if it has a feathering in its [[Stone–Čech compactification|Stone–Čech compactification]] or [[Wallman compactification|Wallman compactification]]. Every complete space (in the sense of Čech) is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832015.png" />-space. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832016.png" />-space has pointwise countable type. In a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832017.png" />-space, the addition theorem for weight holds and the net weight coincides with the weight. Paracompact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832018.png" />-spaces are perfect pre-images of metric spaces. Paracompact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832019.png" />-spaces with a pointwise countable base are metrizable, just as spaces of this type with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832020.png" />-diagonal are also metrizable. The perfect image and the perfect pre-image of a paracompact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832021.png" />-space are also paracompact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832022.png" />-spaces.
 
  
  
  
 
====Comments====
 
====Comments====
The word  "plumingpluming" is also used instead of feathering. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038320/f03832023.png" />-space is also called a [[Feathered space|feathered space]].
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The word  ''pluming'' is also used instead of feathering. A $P$-space is also called a ''[[feathered space]]''.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  "Generalized metric spaces"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set-Theoretic Topology'' , North-Holland  (1984)  pp. 423–501</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  "Generalized metric spaces"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set-Theoretic Topology'' , North-Holland  (1984)  pp. 423–501</TD></TR>
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</table>
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Revision as of 16:16, 22 September 2017

of a space

A countable family $P$ of coverings of a space $X$ by open sets in an ambient space $Y$ such that $$ \bigcap \{ \mathrm{St}_\gamma(x) : \gamma \in P \} \subset X $$ for every point $x \in X$; here $\mathrm{St}_\gamma(x)$ denotes the star of the point $x$ relative to $\gamma$, i.e. the union of all elements of $\gamma$ containing the point $x$.

The concept of a feathering forms the basis of the definition of the so-called $P$-space (in the sense of A.V. Arkhangel'skii). A space $X$ is called a $P$-space if it has a feathering in its Stone–Čech compactification or Wallman compactification. Every complete space (in the sense of Čech) is a $P$-space. Every $P$-space has pointwise countable type. In a $P$-space, the addition theorem for weight holds and the net weight coincides with the weight. Paracompact $P$-spaces are perfect pre-images of metric spaces. Paracompact $P$-spaces with a pointwise countable base are metrizable, just as spaces of this type with a $G_\delta$-diagonal are also metrizable. The perfect image and the perfect pre-image of a paracompact $P$-space are also paracompact $P$-spaces.


Comments

The word pluming is also used instead of feathering. A $P$-space is also called a feathered space.

References

[a1] "Generalized metric spaces" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 423–501
How to Cite This Entry:
Feathering. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feathering&oldid=41923
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article