Feathered space

-space

A completely-regular Hausdorff space having a feathering in some Hausdorff compactification. A feathering in of a subspace of a topological space is a countable system of families of open sets in such that for each point the intersection of its stars with respect to the families over all is contained in and contains the point . Here the star of a point with respect to a family of sets is the union of all elements of containing . If a space has a feathering in some Hausdorff compactification of it, then it has a feathering in every Hausdorff compactification. If a set is the intersection of a sequence of sets open in a space , then the system constitutes a feathering of the subspace in . In particular, if a space is Čech complete, i.e. if it is a -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 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 -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 onto some metric space, it is necessary and sufficient that 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 can be mapped continuously and one-to-one onto a metric space, then is metrizable. On this basis it has been shown that a Tikhonov space is metrizable if and only if it is a paracompact feathered space with diagonal; the latter condition means that the set can be represented as the intersection of a countable family of open sets in . 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 -spaces). They are not to be confused with -spaces, which is a term for various other, inequivalent, notions.

Among paracompact spaces, plumed spaces coincide with the -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