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(→‎-space in the sense of Morita.: cf Morita conjectures)
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==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300202.png" />-space in the sense of Gillman–Henriksen.==
 
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300203.png" />-space as defined in [[#References|[a2]]] is a [[Completely-regular space|completely-regular space]] in which every point is a [[P-point|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300204.png" />-point]], i.e., every fixed prime ideal in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300205.png" /> of real-valued continuous functions is maximal (cf. also [[Maximal ideal|Maximal ideal]]; [[Prime ideal|Prime ideal]]); this is equivalent to saying that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300206.png" />-subset is open (cf. also [[Set of type F sigma(G delta)|Set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300207.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300208.png" />)]]). The latter condition is used to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300209.png" />-spaces among general topological spaces. In [[#References|[a5]]] these spaces were called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002011.png" />-additive, because countable unions of closed sets are closed.
 
  
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).
+
$P$-space or $p$-space refers to various classes of [[topological space]], discussed below.
  
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.
+
[[PSPACE]] or $\mathcal{P}$-space refers to an [[Algorithm, computational complexity of an|algorithmic complexity class]].
  
==<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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==$P$-space in the sense of Gillman–Henriksen.==
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002045.png" />-space as defined in [[#References|[a3]]] is a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002046.png" /> with the following covering property: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002047.png" /> be a set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002048.png" /> be a family of open sets (indexed by the set of finite sequences of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002049.png" />). Then there is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002050.png" /> of closed sets such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002051.png" /> and whenever a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002052.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002053.png" />, then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002054.png" />. K. Morita introduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002055.png" />-spaces to characterize spaces whose products with all metrizable spaces are normal (cf. also [[Normal space|Normal space]]): A space is a normal (paracompact) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002056.png" />-space if and only if its product with every [[Metrizable space|metrizable space]] is normal (paracompact, cf. also [[Paracompact space|Paracompact space]]).
+
A $P$-space as defined in [[#References|[a2]]] is a [[Completely-regular space|completely-regular space]] in which every point is a [[P-point|$P$-point]], ''i.e.'', every fixed prime ideal in the ring $C(X)$ of real-valued continuous functions is maximal (cf. also [[Maximal ideal|Maximal ideal]]; [[Prime ideal|Prime ideal]]); this is equivalent to saying that every $G_\delta$-subset is open (cf. also [[Set of type F sigma(G delta)|Set of type $F_\sigma$ ($G_\delta$)]]). The latter condition is used to define $P$-spaces among general topological spaces. In [[#References|[a5]]] these spaces were called $\aleph_1$-additive, because countable unions of closed sets are closed.
  
Morita [[#References|[a4]]] conjectured that this characterization is symmetric in that a space is metrizable if and only if its product with every normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002057.png" />-space is normal. K. Chiba, T.C. Przymusiński and M.E. Rudin [[#References|[a1]]] showed that the conjecture is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002058.png" />, i.e. Gödel's [[axiom of constructibility]], holds (cf. also [[Gödel constructive set|Gödel constructive set]]). These authors also showed that another conjecture of Morita is true without any extra set-theoretic axioms: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002059.png" /> is normal for every normal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002060.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002061.png" /> is discrete: cf. [[Morita conjectures]].
+
Non-Archimedean ordered fields are $P$-spaces, in their order topology; thus, $P$-spaces occur in non-standard analysis. Another source of $P$-spaces is formed by the $\omega_\mu$-metrizable spaces of [[#References|[a5]]]. If $\omega_\mu$ is a regular cardinal number (cf. also [[Cardinal number|Cardinal number]]), then an $\omega_\mu$-metrizable space is a set $X$ with a mapping $d$ from $X\times X$ to the ordinal $\omega_\mu+1$ that acts like a [[Metric|metric]]: $d(x,y) = \omega_\mu$ if and only if $x=y$; $d(x,y) = d(y,x)$ and $d(x,z) \ge \min\{d(x,y), d(y,z)\}$; $d$ is called an $\omega_\mu$-metric. A topology is formed, as for a [[Metric space|metric space]], using $d$-balls: $B(x,\alpha) = \{y : d(x,y) \ge \alpha\}$, where $\alpha < \omega_\mu$. The $\omega_0$-metrizable spaces are exactly the strongly zero-dimensional metric spaces [[#References|[a8]]] (cf. also [[Zero-dimensional space|Zero-dimensional space]]). If $\omega_\mu$ is uncountable, then $(X,d)$ is a $P$-space (and conversely).
  
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.
+
One also employs $P$-spaces in the investigation of [[box product]]s [[#References|[a7]]]. If a product $X = \prod_{i=1}^\infty X_i$ is endowed with the box topology, then the equivalence relation $x \equiv y$ defined by $\{i : x_i \ne y_i\}$ being finite defines a quotient space of $X$, denoted $\nabla_{i=1}^\infty X_i$, that is a $P$-space. The quotient mapping is open and the box product and its quotient share many properties.
 +
 
 +
==$P$-space in the sense of Morita.==
 +
A $P$-space as defined in [[#References|[a3]]] is a [[Topological space|topological space]] $X$ with the following covering property: Let $\Omega$ be a set and let $\{G(\alpha_1, \dots,\alpha_n):  \alpha_1, \dots,\alpha_n \in \Omega\}$ be a family of open sets (indexed by the set of finite sequences of elements of $\Omega$). Then there is a family $\{F(\alpha_1, \dots,\alpha_n):  \alpha_1, \dots,\alpha_n \in \Omega\}$ of closed sets such that $F(\alpha_1, \dots,\alpha_n) \subseteq G(\alpha_1, \dots,\alpha_n)$ and whenever a sequence $(\alpha_i)_{i=1}^{\infty}$ satisfies $\cup_{n=1}^\infty G(\alpha_1, \ldots, \alpha_n) = X$, then also $\cup_{n=1}^\infty F(\alpha_1, \ldots, \alpha_n) = X$. K. Morita introduced $P$-spaces to characterize spaces whose products with all metrizable spaces are normal (cf. also [[Normal space|Normal space]]): A space is a normal (paracompact) $P$-space if and only if its product with every [[Metrizable space|metrizable space]] is normal (paracompact, cf. also [[Paracompact space|Paracompact space]]).
 +
 
 +
Morita [[#References|[a4]]] conjectured that this characterization is symmetric in that a space is metrizable if and only if its product with every normal $P$-space is normal. K. Chiba, T.C. Przymusiński and M.E. Rudin [[#References|[a1]]] showed that the conjecture is true if $V=L$, ''i.e.'' Gödel's [[axiom of constructibility]], holds (cf. also [[Gödel constructive set|Gödel constructive set]]). These authors also showed that another conjecture of Morita is true without any extra set-theoretic axioms: If $X \times Y$ is normal for every normal space $Y$, then $X$ is discrete: cf. [[Morita conjectures]].
 +
 
 +
There is a characterization of $P$-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 $U_1,U_2,\dots$ and player II chooses closed sets $F_1,F_2,\dots$, with the proviso that $F_n \subseteq \cup_{i\le n} U_i$. Player II wins the play if $\bigcup_n F_n = X$. One can show that Player II has a winning strategy if and only if $X$ is a $P$-space.
 +
 
 +
==$p$-space in the sense of Arkhangel'skii.==
 +
A  [[feathered space]] or plumed space, a completely-regular Hausdorff space having a [[feathering]] in some Hausdorff compactification, has been termed a $p$-space. For [[paracompact space]]s these coincide with the $p$-spaces of Morita, [[#References|[b1]]].
  
 
====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 $P$-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>
 +
 
 +
<TR><TD valign="top">[b1]</TD> <TD valign="top">  J.-I. Nagata,  "Modern general topology" , North-Holland  (1985)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 11:05, 13 February 2024

$P$-space or $p$-space refers to various classes of topological space, discussed below.

PSPACE or $\mathcal{P}$-space refers to an algorithmic complexity class.

$P$-space in the sense of Gillman–Henriksen.

A $P$-space as defined in [a2] is a completely-regular space in which every point is a $P$-point, i.e., every fixed prime ideal in the ring $C(X)$ of real-valued continuous functions is maximal (cf. also Maximal ideal; Prime ideal); this is equivalent to saying that every $G_\delta$-subset is open (cf. also Set of type $F_\sigma$ ($G_\delta$)). The latter condition is used to define $P$-spaces among general topological spaces. In [a5] these spaces were called $\aleph_1$-additive, because countable unions of closed sets are closed.

Non-Archimedean ordered fields are $P$-spaces, in their order topology; thus, $P$-spaces occur in non-standard analysis. Another source of $P$-spaces is formed by the $\omega_\mu$-metrizable spaces of [a5]. If $\omega_\mu$ is a regular cardinal number (cf. also Cardinal number), then an $\omega_\mu$-metrizable space is a set $X$ with a mapping $d$ from $X\times X$ to the ordinal $\omega_\mu+1$ that acts like a metric: $d(x,y) = \omega_\mu$ if and only if $x=y$; $d(x,y) = d(y,x)$ and $d(x,z) \ge \min\{d(x,y), d(y,z)\}$; $d$ is called an $\omega_\mu$-metric. A topology is formed, as for a metric space, using $d$-balls: $B(x,\alpha) = \{y : d(x,y) \ge \alpha\}$, where $\alpha < \omega_\mu$. The $\omega_0$-metrizable spaces are exactly the strongly zero-dimensional metric spaces [a8] (cf. also Zero-dimensional space). If $\omega_\mu$ is uncountable, then $(X,d)$ is a $P$-space (and conversely).

One also employs $P$-spaces in the investigation of box products [a7]. If a product $X = \prod_{i=1}^\infty X_i$ is endowed with the box topology, then the equivalence relation $x \equiv y$ defined by $\{i : x_i \ne y_i\}$ being finite defines a quotient space of $X$, denoted $\nabla_{i=1}^\infty X_i$, that is a $P$-space. The quotient mapping is open and the box product and its quotient share many properties.

$P$-space in the sense of Morita.

A $P$-space as defined in [a3] is a topological space $X$ with the following covering property: Let $\Omega$ be a set and let $\{G(\alpha_1, \dots,\alpha_n):  \alpha_1, \dots,\alpha_n \in \Omega\}$ be a family of open sets (indexed by the set of finite sequences of elements of $\Omega$). Then there is a family $\{F(\alpha_1, \dots,\alpha_n):  \alpha_1, \dots,\alpha_n \in \Omega\}$ of closed sets such that $F(\alpha_1, \dots,\alpha_n) \subseteq G(\alpha_1, \dots,\alpha_n)$ and whenever a sequence $(\alpha_i)_{i=1}^{\infty}$ satisfies $\cup_{n=1}^\infty G(\alpha_1, \ldots, \alpha_n) = X$, then also $\cup_{n=1}^\infty F(\alpha_1, \ldots, \alpha_n) = X$. K. Morita introduced $P$-spaces to characterize spaces whose products with all metrizable spaces are normal (cf. also Normal space): A space is a normal (paracompact) $P$-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 $P$-space is normal. K. Chiba, T.C. Przymusiński and M.E. Rudin [a1] showed that the conjecture is true if $V=L$, 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 $X \times Y$ is normal for every normal space $Y$, then $X$ is discrete: cf. Morita conjectures.

There is a characterization of $P$-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 $U_1,U_2,\dots$ and player II chooses closed sets $F_1,F_2,\dots$, with the proviso that $F_n \subseteq \cup_{i\le n} U_i$. Player II wins the play if $\bigcup_n F_n = X$. One can show that Player II has a winning strategy if and only if $X$ is a $P$-space.

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

A feathered space or plumed space, a completely-regular Hausdorff space having a feathering in some Hausdorff compactification, has been termed a $p$-space. For paracompact spaces these coincide with the $p$-spaces of Morita, [b1].

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 $P$-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
[b1] J.-I. Nagata, "Modern general topology" , North-Holland (1985)
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