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(→‎-space in the sense of Morita.: cf Morita conjectures)
(Add section: P-space in the sense of Arkhangel'skii. Points to feathered space.)
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Non-Archimedean ordered fields are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002012.png" />-spaces, in their order topology; thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002013.png" />-spaces occur in non-standard analysis. Another source of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002014.png" />-spaces is formed by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002015.png" />-metrizable spaces of [[#References|[a5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002016.png" /> is a regular cardinal number (cf. also [[Cardinal number|Cardinal number]]), then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002018.png" />-metrizable space is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002019.png" /> with a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002020.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002021.png" /> to the ordinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002022.png" /> that acts like a [[Metric|metric]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002023.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002024.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002026.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002027.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002029.png" />-metric. A topology is formed, as for a [[Metric space|metric space]], using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002030.png" />-balls: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002032.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002033.png" />-metrizable spaces are exactly the strongly zero-dimensional metric spaces [[#References|[a8]]] (cf. also [[Zero-dimensional space|Zero-dimensional space]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002034.png" /> is uncountable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002035.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002036.png" />-space (and conversely).
 
Non-Archimedean ordered fields are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002012.png" />-spaces, in their order topology; thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002013.png" />-spaces occur in non-standard analysis. Another source of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002014.png" />-spaces is formed by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002015.png" />-metrizable spaces of [[#References|[a5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002016.png" /> is a regular cardinal number (cf. also [[Cardinal number|Cardinal number]]), then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002018.png" />-metrizable space is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002019.png" /> with a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002020.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002021.png" /> to the ordinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002022.png" /> that acts like a [[Metric|metric]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002023.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002024.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002026.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002027.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002029.png" />-metric. A topology is formed, as for a [[Metric space|metric space]], using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002030.png" />-balls: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002032.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002033.png" />-metrizable spaces are exactly the strongly zero-dimensional metric spaces [[#References|[a8]]] (cf. also [[Zero-dimensional space|Zero-dimensional space]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002034.png" /> is uncountable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002035.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002036.png" />-space (and conversely).
  
One also employs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002037.png" />-spaces in the investigation of box products (cf. also [[Topological product|Topological product]]), [[#References|[a7]]]. If a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002038.png" /> is endowed with the box topology, then the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002039.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002040.png" /> is finite and defines a quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002041.png" />, denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002042.png" />, that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002043.png" />-space. The quotient mapping is open and the box product and its quotient share many properties.
+
One also employs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002037.png" />-spaces in the investigation of [[box product]]s [[#References|[a7]]]. If a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002038.png" /> is endowed with the box topology, then the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002039.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002040.png" /> being finite defines a quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002041.png" />, denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002042.png" />, that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002043.png" />-space. The quotient mapping is open and the box product and its quotient share many properties.
  
 
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002044.png" />-space in the sense of Morita.==
 
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002044.png" />-space in the sense of Morita.==
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There is a characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002062.png" />-spaces in terms of topological games [[#References|[a6]]]; let two players, I and II, play the following game on a topological space: player I chooses open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002063.png" /> and player II chooses closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002064.png" />, with the proviso that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002065.png" />. Player II wins the play if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002066.png" />. One can show that Player II has a winning strategy if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002067.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002068.png" />-space.
 
There is a characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002062.png" />-spaces in terms of topological games [[#References|[a6]]]; let two players, I and II, play the following game on a topological space: player I chooses open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002063.png" /> and player II chooses closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002064.png" />, with the proviso that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002065.png" />. Player II wins the play if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002066.png" />. One can show that Player II has a winning strategy if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002067.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002068.png" />-space.
 +
 +
==$P$-space in the sense of Arkhangel'skii.==
 +
A completely-regular Hausdorff space having a [[feathering]] in some Hausdorff compactification: a [[feathered space]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Chiba,  T.C. Przymusiński,  M.E. Rudin,  "Normality of products and Morita's conjectures"  ''Topol. Appl.'' , '''22'''  (1986)  pp. 19–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gillman,  M. Henriksen,  "Concerning rings of continuous functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 340–362</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  K. Morita,  "Some problems on normality of products of spaces"  J. Novák (ed.) , ''Proc. Fourth Prague Topological Symp. (Prague, August 1976)'' , Soc. Czech. Math. and Physicists , Prague  (1977)  pp. 296–297  (Part B: Contributed papers)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Sikorski,  "Remarks on some topological spaces of high power"  ''Fundam. Math.'' , '''37'''  (1950)  pp. 125–136</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Telgárski,  "A characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002069.png" />-spaces"  ''Proc. Japan Acad.'' , '''51'''  (1975)  pp. 802–807</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.W. Williams,  "Box products"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set Theoretic Topology'' , North-Holland  (1984)  pp. Chap. 4; 169–200</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. de Groot,  "Non-Archimedean metrics in topology"  ''Proc. Amer. Math. Soc.'' , '''7'''  (1956)  pp. 948–953</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Chiba,  T.C. Przymusiński,  M.E. Rudin,  "Normality of products and Morita's conjectures"  ''Topol. Appl.'' , '''22'''  (1986)  pp. 19–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gillman,  M. Henriksen,  "Concerning rings of continuous functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 340–362</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  K. Morita,  "Some problems on normality of products of spaces"  J. Novák (ed.) , ''Proc. Fourth Prague Topological Symp. (Prague, August 1976)'' , Soc. Czech. Math. and Physicists , Prague  (1977)  pp. 296–297  (Part B: Contributed papers)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Sikorski,  "Remarks on some topological spaces of high power"  ''Fundam. Math.'' , '''37'''  (1950)  pp. 125–136</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Telgárski,  "A characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002069.png" />-spaces"  ''Proc. Japan Acad.'' , '''51'''  (1975)  pp. 802–807</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.W. Williams,  "Box products"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set Theoretic Topology'' , North-Holland  (1984)  pp. Chap. 4; 169–200</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. de Groot,  "Non-Archimedean metrics in topology"  ''Proc. Amer. Math. Soc.'' , '''7'''  (1956)  pp. 948–953</TD></TR></table>

Revision as of 20:03, 23 September 2017

-space in the sense of Gillman–Henriksen.

A -space as defined in [a2] is a completely-regular space in which every point is a -point, i.e., every fixed prime ideal in the ring of real-valued continuous functions is maximal (cf. also Maximal ideal; Prime ideal); this is equivalent to saying that every -subset is open (cf. also Set of type ()). The latter condition is used to define -spaces among general topological spaces. In [a5] these spaces were called -additive, because countable unions of closed sets are closed.

Non-Archimedean ordered fields are -spaces, in their order topology; thus, -spaces occur in non-standard analysis. Another source of -spaces is formed by the -metrizable spaces of [a5]. If is a regular cardinal number (cf. also Cardinal number), then an -metrizable space is a set with a mapping from to the ordinal that acts like a metric: if and only if ; and ; is called an -metric. A topology is formed, as for a metric space, using -balls: , where . The -metrizable spaces are exactly the strongly zero-dimensional metric spaces [a8] (cf. also Zero-dimensional space). If is uncountable, then is a -space (and conversely).

One also employs -spaces in the investigation of box products [a7]. If a product is endowed with the box topology, then the equivalence relation defined by being finite defines a quotient space of , denoted , that is a -space. The quotient mapping is open and the box product and its quotient share many properties.

-space in the sense of Morita.

A -space as defined in [a3] is a topological space with the following covering property: Let be a set and let be a family of open sets (indexed by the set of finite sequences of elements of ). Then there is a family of closed sets such that and whenever a sequence satisfies , then also . K. Morita introduced -spaces to characterize spaces whose products with all metrizable spaces are normal (cf. also Normal space): A space is a normal (paracompact) -space if and only if its product with every metrizable space is normal (paracompact, cf. also Paracompact space).

Morita [a4] conjectured that this characterization is symmetric in that a space is metrizable if and only if its product with every normal -space is normal. K. Chiba, T.C. Przymusiński and M.E. Rudin [a1] showed that the conjecture is true if , i.e. Gödel's axiom of constructibility, holds (cf. also Gödel constructive set). These authors also showed that another conjecture of Morita is true without any extra set-theoretic axioms: If is normal for every normal space , then is discrete: cf. Morita conjectures.

There is a characterization of -spaces in terms of topological games [a6]; let two players, I and II, play the following game on a topological space: player I chooses open sets and player II chooses closed sets , with the proviso that . Player II wins the play if . One can show that Player II has a winning strategy if and only if is a -space.

$P$-space in the sense of Arkhangel'skii.

A completely-regular Hausdorff space having a feathering in some Hausdorff compactification: a feathered space.

References

[a1] K. Chiba, T.C. Przymusiński, M.E. Rudin, "Normality of products and Morita's conjectures" Topol. Appl. , 22 (1986) pp. 19–32
[a2] L. Gillman, M. Henriksen, "Concerning rings of continuous functions" Trans. Amer. Math. Soc. , 77 (1954) pp. 340–362
[a3] K. Morita, "Products of normal spaces with metric spaces" Math. Ann. , 154 (1964) pp. 365–382
[a4] K. Morita, "Some problems on normality of products of spaces" J. Novák (ed.) , Proc. Fourth Prague Topological Symp. (Prague, August 1976) , Soc. Czech. Math. and Physicists , Prague (1977) pp. 296–297 (Part B: Contributed papers)
[a5] R. Sikorski, "Remarks on some topological spaces of high power" Fundam. Math. , 37 (1950) pp. 125–136
[a6] R. Telgárski, "A characterization of -spaces" Proc. Japan Acad. , 51 (1975) pp. 802–807
[a7] S.W. Williams, "Box products" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set Theoretic Topology , North-Holland (1984) pp. Chap. 4; 169–200
[a8] J. de Groot, "Non-Archimedean metrics in topology" Proc. Amer. Math. Soc. , 7 (1956) pp. 948–953
How to Cite This Entry:
P-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-space&oldid=35013
This article was adapted from an original article by K.P. Hart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article