View source for Feathered space

''$P$-space''

A [[Completely-regular space|completely-regular]] [[Hausdorff space]] having a [[feathering]] in some Hausdorff [[compactification]]. A feathering in $Y$ of a subspace $X$ of a topological space $Y$ is a countable system $\mathcal{P}$ of families of open sets in $Y$ such that for each point $x \in X$ the intersection of its stars $\mathrm{St}_\gamma(x)$ with respect to the families $\gamma$ over all $\gamma \in \mathcal{P}$ is contained in $X$ and contains the point $x$. Here the star $\mathrm{St}_\gamma(x)$ of a point $x$ with respect to a family of sets $\gamma$ is the union of all elements of $\gamma$ containing $x$. If a space $X$ has a feathering in some Hausdorff compactification, then it has a feathering in every Hausdorff compactification. If a set $X$ is the intersection of a sequence $U_1,U_2,\ldots$ of sets open in a space $Y$, then the system $\{ \{U_1\}, \{U_2\}, \ldots\}$ constitutes a feathering of the subspace $X$ in $Y$. In particular, if a space is [[Čech-complete space|Čech complete]], i.e. if it is a [[G-delta|$G_\delta$]]-set in some Hausdorff compactification, then it is a feathered space. All metric spaces are feathered. Therefore, the concept of a feathered space is an extension of both the concept of a locally compact space and the concept of a metric space.

The class of feathered spaces is stable under the formation of countable products and passage to closed or $G_\delta$ subspaces. The pre-image of a feathered space under a [[perfect mapping]] is a feathered space (in the class of Tikhonov spaces). The assumption of a space being feathered guarantees a good behaviour in many important respects. Any feathered space is a $k$-space. A countable feathered space has a countable base. Moreover, if a feathered space contains a countable network, then it has a countable base (and is metrizable). Under a continuous mapping onto a feathered space the weight cannot increase. It is important that the behaviour of certain other fundamental characteristics essentially changes in the presence of a feathering. In particular, a countable product of [[Paracompact space|paracompact]] feathered spaces is a paracompact feathered space, although paracompactness itself is not preserved under taking finite products. Also, a product of countably many finally-compact feathered spaces is a finally-compact feathered space, although final compactness is not preserved under finite products. The concept of a feathering enables one to characterize those spaces that can be mapped perfectly onto metric spaces. That is, for there to be a perfect mapping of a Tikhonov space $X$ onto some metric space, it is necessary and sufficient that $X$ be a paracompact feathered space (Arkhangel'skii's theorem). The image of a paracompact feathered space under a perfect mapping is a paracompact feathered space (Filippov's theorem); however, an example is known of a perfect mapping of a feathered space onto a non-feathered Tikhonov space. Important examples of non-paracompact feathered spaces are provided by the non-paracompact locally compact spaces and by the non-metrizable Moore spaces — Tikhonov spaces with countable developments. Paracompactness follows from being feathered for the space of a topological group. A simple criterion for being feathered applies for groups: The space of a topological group is feathered if and only if it contains a non-empty Hausdorff compactum having a countable defining system of neighbourhoods (Pasynkov's theorem). In the presence of a feathering, the metrizability criteria simplify considerably. In particular, if a paracompact feathered space $X$ can be mapped continuously and one-to-one onto a metric space, then $X$ is metrizable. On this basis it has been shown that a Tikhonov space $X$ is metrizable if and only if it is a paracompact feathered space with $G_\delta$ diagonal; the latter condition means that the set $\Delta = \{(x,x) : x \in X\}$ can be represented as the intersection of a countable family of open sets in $X \times X$. These results and others enable one to consider the property of being feathered as one of the basic general properties of metric spaces and Hausdorff compacta, along with paracompactness.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   "A class of spaces which contains all metric and all locally compact spaces"  ''Mat. Sb.'' , '''67''' :  1  (1965)  pp. 55–88  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Filippov,   "The perfect image of a paracompact feathered space"  ''Soviet Math. Dokl.'' , '''8'''  (1967)  pp. 1151–1153  ''Dokl. Akad. Nauk SSSR'' , '''176''' :  3  (1967)  pp. 533–535</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Pasynkov,   "ALmost-metrizable topological groups"  ''Soviet Math. Dokl.'' , '''7'''  (1966)  pp. 404–408  ''Dokl. Akad. Nauk SSSR'' , '''161''' :  2  (1965)  pp. 281–284</TD></TR>
</table>



====Comments====
In the English literature, a feathering is also called a pluming (see also [[Feathering]]), hence feathered spaces are also called plumed spaces (abbreviated to $P$-spaces). They are not to be confused with various other, inequivalent, notions of [[P-space|$P$-space]].

Among paracompact spaces, plumed spaces coincide with the [[P-space#-space_in_the_sense_of_Morita.|$P$-spaces]] introduced by K. Morita [[#References|[a1]]], but in the absence of paracompactness the two definitions are not equivalent. For more details, see [[#References|[a2]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Morita,   "Products of normal spaces with metric spaces"  ''Math. Ann.'' , '''154'''  (1964)  pp. 365–382</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-I. Nagata,   "Modern general topology" , North-Holland  (1985)</TD></TR>
<TR><TD valign="top">[a3]</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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