# 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.

## $P$-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) |

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P-space.

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