# P-space

$P$-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].

How to Cite This Entry:
P-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-space&oldid=52978
This article was adapted from an original article by K.P. Hart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article