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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383102.png" />-space''
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''$p$-space''
  
A completely-regular Hausdorff space having a feathering in some Hausdorff compactification. A feathering in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383103.png" /> of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383104.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383105.png" /> is a countable system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383106.png" /> of families of open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383107.png" /> such that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383108.png" /> the intersection of its stars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f0383109.png" /> with respect to the families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831010.png" /> over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831011.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831012.png" /> and contains the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831013.png" />. Here the star <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831014.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831015.png" /> with respect to a family of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831016.png" /> is the union of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831017.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831018.png" />. If a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831019.png" /> has a feathering in some Hausdorff [[Compactification|compactification]] of it, then it has a feathering in every Hausdorff compactification. If a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831020.png" /> is the intersection of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831021.png" /> of sets open in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831022.png" />, then the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831023.png" /> constitutes a feathering of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831025.png" />. In particular, if a space is Čech complete, i.e. if it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831026.png" />-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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831028.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831029.png" /> onto some metric space, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831031.png" /> can be mapped continuously and one-to-one onto a metric space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831032.png" /> is metrizable. On this basis it has been shown that a Tikhonov space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831033.png" /> is metrizable if and only if it is a paracompact feathered space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831034.png" /> diagonal; the latter condition means that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831035.png" /> can be represented as the intersection of a countable family of open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831036.png" />. 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.
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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====
 
====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>
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<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====
 
====Comments====
In the English literature, a feathering is also called a pluming (see also [[Feathering|Feathering]]), hence feathered spaces are also called plumed spaces (abbreviated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831038.png" />-spaces). They are not to be confused with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831039.png" />-spaces, which is a term for various other, inequivalent, notions.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038310/f03831041.png" />-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]]].
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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====
 
====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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<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>
 +
 
 +
{{TEX|done}}

Latest revision as of 20:32, 23 September 2017

$p$-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
How to Cite This Entry:
Feathered space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Feathered_space&oldid=11456
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article