Difference between revisions of "P-space"
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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. | 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. | ||
− | == | + | ==$P$-space in the sense of Morita.== |
− | A | + | 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 <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 $P$-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 | + | 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 | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002065.png" />. 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.== | ==$p$-space in the sense of Arkhangel'skii.== |
Revision as of 12:49, 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 -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.
$P$-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