The projective character bounds the order of a base
Authors:
István Juhász and Zoltán Szentmiklóssy
Journal:
Proc. Amer. Math. Soc. 136 (2008), 29792984
MSC (2000):
Primary 54A25, 54C10, 54D70
Published electronically:
April 2, 2008
MathSciNet review:
2399066
Fulltext PDF Free Access
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Abstract: All spaces below are Tychonov. We define the projective  character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a base whose order is ; that is, every point in is contained in at most many members of the base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.
 1.
A.
V. Arhangel′skiĭ, The structure and classification of
topological spaces and cardinal invariants, Uspekhi Mat. Nauk
33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
(80i:54005)
 2.
A. V. Arhangel'skii, Precalibers, monolithic spaces, first countability, and homogeneity in the class of compact spaces, preprint.
 3.
István
Juhász, Cardinal functions in topology—ten years
later, 2nd ed., Mathematical Centre Tracts, vol. 123,
Mathematisch Centrum, Amsterdam, 1980. MR 576927
(82a:54002)
 4.
István
Juhász, Lajos
Soukup, and Zoltán
Szentmiklóssy, First countable spaces without
pointcountable 𝜋bases, Fund. Math. 196
(2007), no. 2, 139–149. MR 2342624
(2008i:54008), http://dx.doi.org/10.4064/fm19624
 5.
B. E. Shapirovskii, On the tightness, weight and related notions, Uch. Zap. Latv. Univ. 257 (1976), pp. 8899.
 6.
B.
È. Shapirovskiĭ, Special types of embeddings in
Tychonoff cubes. Subspaces of Σproducts and cardinal
invariants, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978)
Colloq. Math. Soc. János Bolyai, vol. 23, NorthHolland,
AmsterdamNew York, 1980, pp. 1055–1086. MR 588855
(82d:54010)
 7.
B.
È. Šapirovskiĭ, Mappings on Tihonov
cubes, Uspekhi Mat. Nauk 35 (1980), no. 3(213),
122–130 (Russian). International Topology Conference (Moscow State
Univ., Moscow, 1979). MR 580628
(82d:54018)
 8.
B. E. Shapirovskii, Cardinal invariants in compact Hausdorff spaces, Amer. Math. Soc. Transl. 134 (1987), pp. 93118.
 9.
Stevo
Todorčević, Free sequences, Topology Appl.
35 (1990), no. 23, 235–238. MR 1058803
(91f:54003), http://dx.doi.org/10.1016/01668641(90)90108E
 1.
 A. V. Arhangel'skii, Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys 33 (1978), pp. 2984. MR 526012 (80i:54005)
 2.
 A. V. Arhangel'skii, Precalibers, monolithic spaces, first countability, and homogeneity in the class of compact spaces, preprint.
 3.
 I. Juhász, Cardinal functions  10 years later, Math. Center Tract no. 123, Amsterdam, 1980. MR 576927 (82a:54002)
 4.
 I. Juhász, L. Soukup, and Z. Szentmiklóssy, First countable spaces without pointcountable bases, Fund. Math., 196 (2007), pp. 139149. MR 2342624
 5.
 B. E. Shapirovskii, On the tightness, weight and related notions, Uch. Zap. Latv. Univ. 257 (1976), pp. 8899.
 6.
 B. E. Shapirovskii, Special types of embeddings in Tychonoff cubes. Subspaces of products and cardinal invariants, in: Topology, Coll. Math. Soc. J. Bolyai 23 (NorthHolland, Amsterdam, 1980), pp. 10551086. MR 588855 (82d:54010)
 7.
 B. E. Shapirovskii, Mappings on Tikhonov cubes, Russian Math. Surveys 35 (1980), pp. 145156 (1981). MR 580628 (82d:54018)
 8.
 B. E. Shapirovskii, Cardinal invariants in compact Hausdorff spaces, Amer. Math. Soc. Transl. 134 (1987), pp. 93118.
 9.
 S. Todorčević, Free sequences, Top. Appl. 35 (1990), pp. 235238. MR 1058803 (91f:54003)
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Additional Information
István Juhász
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest, H1364 Hungary
Email:
juhasz@renyi.hu
Zoltán Szentmiklóssy
Affiliation:
Department of Analysis, Eötvös Loránt University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
Email:
zoli@renyi.hu
DOI:
http://dx.doi.org/10.1090/S0002993908093155
PII:
S 00029939(08)093155
Keywords:
Projective $\pi $character,
order of a $\pi $base,
irreducible map
Received by editor(s):
March 28, 2007
Received by editor(s) in revised form:
June 17, 2007
Published electronically:
April 2, 2008
Additional Notes:
This research was supported by OTKA grant no. 61600.
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
