P-space

-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 (cf. also Topological product), [a7]. If a product is endowed with the box topology, then the equivalence relation defined by is finite and 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.

-space in the sense of Morita.

A -space as defined in [a3] is a topological space with the following covering property: Let be a set and let be a family of open sets (indexed by the set of finite sequences of elements of ). Then there is a family of closed sets such that and whenever a sequence satisfies , then also . K. Morita introduced -spaces to characterize spaces whose products with all metrizable spaces are normal (cf. also Normal space): A space is a normal (paracompact) -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 -space is normal. K. Chiba, T.C. Przymusiński and M.E. Rudin [a1] showed that the conjecture is true if , 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 is normal for every normal space , then is discrete.

There is a characterization of -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 and player II chooses closed sets , with the proviso that . Player II wins the play if . One can show that Player II has a winning strategy if and only if is a -space.

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 -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