Nonnormal spaces with countable extent

Authors:
Winfried Just, Ol'ga V. Sipacheva and Paul J. Szeptycki

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1227-1235

MSC (1991):
Primary 03E75, 54A20, 54A35, 54C35, 54G20

MathSciNet review:
1343704

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Abstract: Examples of spaces are constructed for which is not normal but all closed discrete subsets are countable. A monolithic example is constructed in ZFC and a separable first countable example is constructed using .

**[A1]**A. V. Arkhangel′skiĭ,*Topological function spaces*, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR**1144519****[A2]**A.V. Arkhangel'skii,*theory*, In: Recent Progress in General Topology, Ed. M. Hu\v{s}ek and J. van Mill, North Holland (1992) 1-56. CMP**93:15****[Ba]**D. P. Baturov,*Subspaces of function spaces*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**4**(1987), 66–69 (Russian). MR**913076****[vD]**Eric K. van Douwen,*The integers and topology*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR**776622****[D]**Alan Dow,*An introduction to applications of elementary submodels to topology*, Topology Proc.**13**(1988), no. 1, 17–72. MR**1031969****[G]**S. P. Gul′ko,*Properties of sets that lie in Σ-products*, Dokl. Akad. Nauk SSSR**237**(1977), no. 3, 505–508 (Russian). MR**0461410****[K]**J. L. Agudin and A. M. Platzeck,*Resolution of the fields of an accelerated charge into their bradyonic and tachyonic parts*, Phys. Lett. A**83**(1981), no. 9, 423–427. MR**621433**, 10.1016/0375-9601(81)90470-9**[L]**N. N. Luzin,*On subsets of the series of natural numbers*, Izvestiya Akad. Nauk SSSR. Ser. Mat.**11**(1947), 403–410 (Russian). MR**0021576****[P]**Teodor C. Przymusiński,*Products of normal spaces*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 781–826. MR**776637****[R]**E. A. Reznichenko,*Normality and collective normality of function spaces*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**6**(1990), 56–58 (Russian); English transl., Moscow Univ. Math. Bull.**45**(1990), no. 6, 25–26. MR**1095997****[Ru]**M. E. Rudin,*-products of metric spaces are normal*, Preprint 1977.

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Additional Information

**Winfried Just**

Affiliation:
Department of Mathematics, Ohio University, Athens, Ohio 45701

Email:
just@ace.cs.ohiou.edu

**Ol'ga V. Sipacheva**

Affiliation:
Chair of General Topology and Geometry, Mechanics and Mathematics Faculty, Moscow State University, 119899 Moscow, Russia

Email:
sipa@glas.apc.org

**Paul J. Szeptycki**

Affiliation:
Department of Mathematics, Ohio University, Athens, Ohio 45701

Email:
szeptyck@ace.cs.ohiou.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03500-9

Keywords:
$C_{p}(X)$,
extent,
normality,
$\diamondsuit$,
almost disjoint family,
$\Psi$-space,
p-ultrafilter,
Luzin gap

Received by editor(s):
April 6, 1994

Additional Notes:
The first author was partially supported by NSF grant DMS-9312363

The second author collaborated while visiting Ohio University

Article copyright:
© Copyright 1996
American Mathematical Society