View source for Feathering

''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

<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>

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====
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]].

====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>