Feathered space

$P$-space

A 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, i.e. if it is a $G_\delta$-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 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

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2]	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)
[3]	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
[4]	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

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.

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

References

[a1]	K. Morita, "Products of normal spaces with metric spaces" Math. Ann. , 154 (1964) pp. 365–382
[a2]	J.-I. Nagata, "Modern general topology" , North-Holland (1985)
[a3]	"Generalized metric spaces" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 423–501