Feathering

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 network 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