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quasi $P$-space; $P$-space; scattered space; Cantor-Bendixson derivatives; \newline nodec space; quasinormality
Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C(X)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset is a $z$-embedded $P$-space, then $X$ is a quasi $P$-space. Conversely, if $X$ is a quasi $P$-space and $F$ is a nowhere dense $z$-embedded zeroset, then $F$ is a $P$-space. On the other hand, there are examples of countable quasi $P$-spaces with no $P$-points at all. If a product $X\times Y$ is normal and quasi $P$, then one of the factors must be a $P$-space. Conversely, if one of the factors is a compact quasi $P$-space and the other a $P$-space then the product is quasi $P$. If $X$ is normal and $X$ and $Y$ are cozero-complemented spaces and $f:X\longrightarrow Y$ is a closed continuous surjection which has the property that $f^{-1}(Z)$ is nowhere dense for each nowhere dense zeroset $Z$, then if $X$ is quasi $P$, so is $Y$. The converse fails even with more stringent assumptions on the map $f$. The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi $P$-spaces is always quasi $P$.
[BSV81] Balcar B., Simon P., Vojtáš P.: Refinement properties and extensions of filters in boolean algebras. Trans. Amer. Math. Soc. 267 (1981), 265-283. DOI 10.1090/S0002-9947-1981-0621987-0 | MR 0621987
[BH87] Ball R.N., Hager A.W.: Archimedean kernel-distinguishing extensions of archimedean $\ell$-groups with weak unit. Indian J. Math. 29 (3) (1987), 351-368. MR 0971646
[BKW77] Bigard A., Keimel K., Wolfenstein S.: Groupes et Anneaux Réticulés. Lecture Notes in Math. 608, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0552653 | Zbl 0384.06022
[Bl76] Blair R.L.: Spaces in which special sets are $z$-embedded. Canad. J. Math. 28 (1976), 673-690. DOI 10.4153/CJM-1976-068-9 | MR 0420542 | Zbl 0359.54009
[Bu80] Burke D.: Closed Mappings. Surveys in Gen. Topology, Academic Press, New York, 1980, pp.1-32. MR 0564098 | Zbl 0476.54017
[CH70] Comfort W., Hager A.: Estimates for the number of continuous functions. Trans. Amer. Math. Soc. 150 (1970), 619-631. DOI 10.1090/S0002-9947-1970-0263016-X | MR 0263016
[CM90] Conrad P., Martinez J.: Complemented lattice-ordered groups. Indag. Math. (N.S.) 1 (1990), 281-297. DOI 10.1016/0019-3577(90)90019-J | MR 1075880 | Zbl 0735.06006
[D95] Darnel M.: Theory of Lattice-Ordered Groups. Pure & Appl. Math. 187, Marcel Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[DF99] Dummit D.S., Foote R.M.: Abstract Algebra. 2nd edition, Prentice Hall, 1999. MR 1138725 | Zbl 1037.00003
[vD93] van Douwen E.: Applications of maximal topologies. Topology Appl. 51 (1993), 125-139. DOI 10.1016/0166-8641(93)90145-4 | MR 1229708 | Zbl 0845.54028
[vDP79] van Douwen E., Pryzmusiński T.: First countable and countable spaces all compactifications of which contain $\beta \Bbb N$, Fund. Math. 52 (1979), 229-234. DOI 10.4064/fm-102-3-229-234 | MR 0532957
[En89] Engelking R.: General Topology. Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[GJ60] Gillman L., Jerison M.: Quotient fields of residue class rings of function rings. Illinois J. Math. 4 (1960), 425-436. DOI 10.1215/ijm/1255456059 | MR 0124727 | Zbl 0098.30701
[GJ76] Gillman L., Jerison M.: Rings of Continuous Functions. Grad. Texts Math. 43, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 0407579 | Zbl 0327.46040
[HM93] Hager A., Martinez J.: Fraction-dense algebras and spaces. Canad. J. Math. 45 (1993), 977-996. DOI 10.4153/CJM-1993-054-6 | MR 1239910 | Zbl 0795.06017
[HJ65] Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110-130. DOI 10.1090/S0002-9947-1965-0194880-9 | MR 0194880 | Zbl 0147.29105
[HLMW94] Henriksen M., Larson S., Martinez J., Woods R.G.: Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. 345 1 (September 1994), 195-221. DOI 10.1090/S0002-9947-1994-1239640-0 | MR 1239640 | Zbl 0817.06014
[HVW87] Henriksen M., Vermeer J., Woods R.G.: Quasi-$F$ covers of Tychonoff spaces. Trans. Amer. Math. Soc. 303 (2) (1987), 779-803. MR 0902798 | Zbl 0653.54025
[Ki01] Kimber C.: $m$-Quasinormal $f$-rings. J. Pure Appl. Algebra 158 (2001), 197-223. DOI 10.1016/S0022-4049(00)00061-X | MR 1822841 | Zbl 0987.06017
[Ko89] Koppelberg S.: Handbook of Boolean Algebras, I. J.D. Monk, Ed., with R. Bonnet; Elsevier, Amsterdam-New York-Oxford-Tokyo, 1989. MR 0991565
[La95] Larson S.: A characterization of $f$-rings in which the sum of semiprime $\ell$-ideals is semiprime and its consequences. Comm. Algebra 23 (1995), 14 5461-5481. DOI 10.1080/00927879508825545 | MR 1363616 | Zbl 0847.06007
[La97a] Larson S.: Quasi-normal $f$-rings. in Proc. Ord. Alg. Structures (Curaçao, 1995), W.C. Holland & J. Martinez, Eds., Kluwer Acad. Publ., Dordrecht, 1997, pp.261-275. MR 1445116 | Zbl 0872.06013
[La97b] Larson S.: $f$-Rings in which every maximal ideal contains finitely many minimal prime ideals. Comm. Algebra 25 (1997), 3859-3888. DOI 10.1080/00927879708826092 | MR 1481572 | Zbl 0952.06026
[Le77] Levy R.: Almost $P$-spaces. Canad. J. Math. 29 (1977), 284-288. DOI 10.4153/CJM-1977-030-7 | MR 0464203 | Zbl 0342.54032
[LR81] Levy R., Rice M.: Normal spaces and the $G_{\delta}$-topology. Colloq. Math. 44 (1981), 227-240. DOI 10.4064/cm-44-2-227-240 | MR 0652582
[Mn69] Mandelker M.: $F'$-spaces and $z$-embedded subspaces. Pacific J. Math. 28 (1969), 615-621. DOI 10.2140/pjm.1969.28.615 | MR 0240782 | Zbl 0172.47903
[MR69] Mioduszewski J., Rudolph L.: $H$-closed and extremally disconnected Hausdorff spaces. Dissertationes Math. LXVI (1969, Warsaw).
[Mo70] Montgomery R.: Structures determined by prime ideals of rings of functions. Trans. Amer. Math. Soc. 147 (1970), 367-380. DOI 10.1090/S0002-9947-1970-0256174-4 | MR 0256174 | Zbl 0222.54014
[Mo73] Montgomery R.: The mapping of prime $z$-ideals. Symp. Math. 17 (1973), 113-124. MR 0440495
[Mr70] Mrowka S.: Some comments on the author's example of a non-R-compact space. Bull. Acad. Polon. Sci., Ser. Math. Astronom. Phys. 18 (1970), 443-448. MR 0268852
[O80] Oxtoby J.: Measure and Category. 2nd edition, Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR 0584443 | Zbl 0435.28011
[PW88] Porter J., Woods R.g.: Extensions and Absolutes of Hausdorff Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1988. MR 0918341 | Zbl 0652.54016
[vRS82] van Rooij A., Shikof W.: A Second Course in Real Analysis. Cambridge Univ. Press, Cambridge, England, 1982.
[Se59] Semadeni Z.: Sur les ensembles clairsemés. Rozprawy Mat. 19 (1959), Warsaw. MR 0107849 | Zbl 0137.16002
[Se71] Semadeni Z.: Banach Spaces of Continuous Functions. Polish Scientific Publishers, Warsaw, 1971. MR 0296671 | Zbl 0478.46014
[T68] Telgársky R.: Total paracompactness and paracompact dispersed spaces. Bull. Acad. Polon. Sci. 16 (1968), 567-572. MR 0235517
[Ve73] Veksler A.G.: $P'$-points, $P'$-sets and $P'$-spaces. A new class of order-continuous measures and functionals. Soviet Math. Dokl. 14 (1973), 1440-1445. MR 0341447
[W75] Weir M.: Hewitt-Nachbin Spaces. North Holland Publ. Co., Amsterdam, 1975. MR 0514909 | Zbl 0314.54002
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