View source for P-space

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300202.png" />-space in the sense of Gillman–Henriksen.==
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300203.png" />-space as defined in [[#References|[a2]]] is a [[Completely-regular space|completely-regular space]] in which every point is a [[P-point|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300204.png" />-point]], i.e., every fixed prime ideal in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p1300205.png" /> 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).

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 products (cf. also [[Topological product|Topological product]]), [[#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" /> is finite and 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.

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002044.png" />-space in the sense of Morita.==
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002045.png" />-space as defined in [[#References|[a3]]] is a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002046.png" /> with the following covering property: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002047.png" /> be a set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002048.png" /> be a family of open sets (indexed by the set of finite sequences of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002049.png" />). Then there is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002050.png" /> of closed sets such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002051.png" /> and whenever a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002052.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002055.png" />-spaces to characterize spaces whose products with all metrizable spaces are normal (cf. also [[Normal space|Normal space]]): A space is a normal (paracompact) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002056.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002057.png" />-space is normal. K. Chiba, T.C. Przymusiński and M.E. Rudin [[#References|[a1]]] showed that the conjecture is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002058.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002059.png" /> is normal for every normal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002060.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002061.png" /> is discrete.

There is a characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002062.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002063.png" /> and player II chooses closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002064.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002066.png" />. One can show that Player II has a winning strategy if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002067.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002068.png" />-space.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Chiba,   T.C. Przymusiński,   M.E. Rudin,   "Normality of products and Morita's conjectures"  ''Topol. Appl.'' , '''22'''  (1986)  pp. 19–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gillman,   M. Henriksen,   "Concerning rings of continuous functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 340–362</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Morita,   "Products of normal spaces with metric spaces"  ''Math. Ann.'' , '''154'''  (1964)  pp. 365–382</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Sikorski,   "Remarks on some topological spaces of high power"  ''Fundam. Math.'' , '''37'''  (1950)  pp. 125–136</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Telgárski,   "A characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130020/p13002069.png" />-spaces"  ''Proc. Japan Acad.'' , '''51'''  (1975)  pp. 802–807</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. de Groot,   "Non-Archimedean metrics in topology"  ''Proc. Amer. Math. Soc.'' , '''7'''  (1956)  pp. 948–953</TD></TR></table>