Difference between revisions of "P-space"
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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.== | ==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300202.png" />-space in the sense of Gillman–Henriksen.== | ||
− | A | + | 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 <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). | 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). |
Revision as of 12:50, 19 March 2023
$P$-space or $p$-space refers to various classes of topological space, discussed below.
PSPACE or $\mathcal{P}$-space refers to an algorithmic complexity class.
-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 -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.
$P$-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 , then also . 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 . 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) |
P-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-space&oldid=52977