## Separation and coding

HTML articles powered by AMS MathViewer

- by Stephen Watson PDF
- Trans. Amer. Math. Soc.
**342**(1994), 83-106 Request permission

## Abstract:

We construct a normal collectionwise Hausdorff space which is not collectionwise normal with respect to copies of [0,1]. We do this by developing a general theory of coding properties into topological spaces. We construct a para-Lindelöf regular space in which para-Lindelöf is coded directly rather than $\sigma$-para-Lindelöf and normal. We construct a normal collectionwise Hausdorff space which is not collectionwise normal in which collectionwise Hausdorff is coded directly rather than obtained as a side-effect to countable approximation. We also show that the Martin’s axiom example of a normal space which is not collectionwise Hausdorff is really just a kind of "dual" of Bing’s space.## References

- Zoltán T. Balogh,
*Paracompactness in locally Lindelöf spaces*, Canad. J. Math.**38**(1986), no. 3, 719–727. MR**845674**, DOI 10.4153/CJM-1986-037-7 - R. H. Bing,
*Metrization of topological spaces*, Canad. J. Math.**3**(1951), 175–186. MR**43449**, DOI 10.4153/cjm-1951-022-3 - R. H. Bing,
*A translation of the normal Moore space conjecture*, Proc. Amer. Math. Soc.**16**(1965), 612–619. MR**181976**, DOI 10.1090/S0002-9939-1965-0181976-6 - Dennis K. Burke,
*Covering properties*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 347–422. MR**776628** - F. S. Cater, Paul Erdős, and Fred Galvin,
*On the density of $\lambda$-box products*, General Topology Appl.**9**(1978), no. 3, 307–312. MR**510912** - Alan Dow, Franklin D. Tall, and William A. R. Weiss,
*New proofs of the consistency of the normal Moore space conjecture. I*, Topology Appl.**37**(1990), no. 1, 33–51. MR**1075372**, DOI 10.1016/0166-8641(90)90013-R - Ryszard Engelking,
*Topologia ogólna*, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR**0500779** - William G. Fleissner,
*The character of $\omega _{1}$ in first countable spaces*, Proc. Amer. Math. Soc.**62**(1976), no. 1, 149–155 (1977). MR**438272**, DOI 10.1090/S0002-9939-1977-0438272-7 - William G. Fleissner,
*A collectionwise Hausdorff nonnormal Moore space with a $\sigma$-locally countable base*, Topology Proc.**4**(1979), no. 1, 83–97 (1980). MR**583690** - William G. Fleissner,
*If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal*, Trans. Amer. Math. Soc.**273**(1982), no. 1, 365–373. MR**664048**, DOI 10.1090/S0002-9947-1982-0664048-8 - William G. Fleissner,
*A normal collectionwise Hausdorff, not collectionwise normal space*, General Topology and Appl.**6**(1976), no. 1, 57–64. MR**391032** - William Fleissner,
*Normal Moore spaces in the constructible universe*, Proc. Amer. Math. Soc.**46**(1974), 294–298. MR**362240**, DOI 10.1090/S0002-9939-1974-0362240-4 - William G. Fleissner,
*Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals*, Proc. Nat. Acad. Sci. U.S.A.**79**(1982), no. 4, 1371–1372. MR**648069**, DOI 10.1073/pnas.79.4.1371
—, - William G. Fleissner and George M. Reed,
*Para-Lindelöf spaces and spaces with a $\sigma$-locally countable base*, Topology Proc.**2**(1977), no. 1, 89–110 (1978). MR**540598** - G. Grjunhage,
*Paracompactness and subparacompactness in perfectly normal locally bicompact spaces*, Uspekhi Mat. Nauk**35**(1980), no. 3(213), 44–49 (Russian). International Topology Conference (Moscow State Univ., Moscow, 1979); Translated from the English by A. V. Arhangel′skiĭ. MR**580619** - Thomas Jech and Karel Prikry,
*Cofinality of the partial ordering of functions from $\omega _{1}$ into $\omega$ under eventual domination*, Math. Proc. Cambridge Philos. Soc.**95**(1984), no. 1, 25–32. MR**727077**, DOI 10.1017/S0305004100061272 - Richard Laver,
*On the consistency of Borel’s conjecture*, Acta Math.**137**(1976), no. 3-4, 151–169. MR**422027**, DOI 10.1007/BF02392416 - Ernest Michael,
*Point-finite and locally finite coverings*, Canadian J. Math.**7**(1955), 275–279. MR**70147**, DOI 10.4153/CJM-1955-029-6
C. Navy, - A. J. Ostaszewski,
*On countably compact, perfectly normal spaces*, J. London Math. Soc. (2)**14**(1976), no. 3, 505–516. MR**438292**, DOI 10.1112/jlms/s2-14.3.505 - G. M. Reed,
*Collectionwise Hausdorff versus collectionwise normal with respect to compact sets*, Topology Appl.**16**(1983), no. 3, 259–272. MR**722119**, DOI 10.1016/0166-8641(83)90023-8 - Mary Ellen Rudin and Michael Starbird,
*Some examples of normal Moore spaces*, Canadian J. Math.**29**(1977), no. 1, 84–92. MR**448311**, DOI 10.4153/CJM-1977-008-9
J. Steprāns, - Franklin D. Tall,
*The density topology*, Pacific J. Math.**62**(1976), no. 1, 275–284. MR**419709** - Franklin D. Tall,
*Normality versus collectionwise normality*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 685–732. MR**776634** - Franklin D. Tall,
*On the existence of normal metacompact Moore spaces which are not metrizable*, Canadian J. Math.**26**(1974), 1–6. MR**377823**, DOI 10.4153/CJM-1974-001-8 - F. D. Tall,
*Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems*, Dissertationes Math. (Rozprawy Mat.)**148**(1977), 53. MR**454913** - Paul Urysohn,
*Über die Mächtigkeit der zusammenhängenden Mengen*, Math. Ann.**94**(1925), no. 1, 262–295 (German). MR**1512258**, DOI 10.1007/BF01208659 - Stephen Watson,
*The character of Bing’s space*, Topology Appl.**28**(1988), no. 2, 171–175. Special issue on set-theoretic topology. MR**932982**, DOI 10.1016/0166-8641(88)90009-0 - Stephen Watson,
*Comments on separation*, Topology Proc.**14**(1989), no. 2, 315–372. MR**1107730** - Stephen Watson,
*A construction of a Dowker space*, Proc. Amer. Math. Soc.**109**(1990), no. 3, 835–841. MR**1019285**, DOI 10.1090/S0002-9939-1990-1019285-X - Stephen Watson,
*Number versus size*, Proc. Amer. Math. Soc.**102**(1988), no. 3, 761–764. MR**929017**, DOI 10.1090/S0002-9939-1988-0929017-X
N. H. Williams,

*Son of George and*$V = L$, J. Symbolic Logic

**48**(1982), 71-77.

*A paralindelöf space which is not paracompact*, Ph.D. thesis, Univ. of Wisconsin-Madison, 1981. P. J. Nyikos,

*Some normal Moore spaces*, Colloq. Math. Soc. János Bolyai

**23**(1978), 883ff.

*Some results in set theory*, Ph.D. thesis, Univ. of Toronto, 1982.

*Combinatorial set theory*, North-Holland, Amsterdam, 1977.

## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**342**(1994), 83-106 - MSC: Primary 54D15; Secondary 54E30, 54G15, 54G20
- DOI: https://doi.org/10.1090/S0002-9947-1994-1225576-8
- MathSciNet review: 1225576