Feathering
of a space
A countable family of coverings of a space
by open sets in an ambient space
such that
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for every point (here
denotes the star of the point
relative to
, i.e. the union of all elements of
containing the point
).
The concept of a feathering forms the basis of the definition of the so-called -space (in the sense of A.V. Arkhangel'skii). A space
is called a
-space if it has a feathering in its Stone–Čech compactification or Wallman compactification. Every complete space (in the sense of Čech) is a
-space. Every
-space has pointwise countable type. In a
-space, the addition theorem for weight holds and the net weight coincides with the weight. Paracompact
-spaces are perfect pre-images of metric spaces. Paracompact
-spaces with a pointwise countable base are metrizable, just as spaces of this type with a
-diagonal are also metrizable. The perfect image and the perfect pre-image of a paracompact
-space are also paracompact
-spaces.
Comments
The word "plumingpluming" is also used instead of feathering. A -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 |
Feathering. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feathering&oldid=41923