Tightness and π-character in centered spaces

Murray Bell

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 2, page 297-307
  • ISSN: 0010-1354

Abstract

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We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = supκ : 2 κ ⊂ X. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered property is not preserved by passage to a zeroset.

How to cite

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Bell, Murray. "Tightness and π-character in centered spaces." Colloquium Mathematicae 80.2 (1999): 297-307. <http://eudml.org/doc/210720>.

@article{Bell1999,
abstract = {We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = supκ : $2^κ$ ⊂ X. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered property is not preserved by passage to a zeroset.},
author = {Bell, Murray},
journal = {Colloquium Mathematicae},
keywords = {centered; tightness; compact; π-character; adequate compact space; centered family; dyadic space},
language = {eng},
number = {2},
pages = {297-307},
title = {Tightness and π-character in centered spaces},
url = {http://eudml.org/doc/210720},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Bell, Murray
TI - Tightness and π-character in centered spaces
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 2
SP - 297
EP - 307
AB - We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = supκ : $2^κ$ ⊂ X. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered property is not preserved by passage to a zeroset.
LA - eng
KW - centered; tightness; compact; π-character; adequate compact space; centered family; dyadic space
UR - http://eudml.org/doc/210720
ER -

References

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  1. [1] A. Arkhangel'skiĭ, Approximation of the theory of dyadic bicompacta, Soviet Math. Dokl. 10 (1969), 151-154. 
  2. [2] A. Arkhangel'skiĭ, On bicompacta hereditarily satisfying Suslin's condition. Tightness and free sequences, ibid. 12 (1971), 1253-1257. Zbl0235.54006
  3. [3] A. Arkhangel'skiĭ, Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys 33 (1978), no. 6, 33-96. Zbl0428.54002
  4. [4] M. Bell, Generalized dyadic spaces, Fund. Math. 125 (1985), 47-58. Zbl0589.54019
  5. [5] M. Bell, G κ subspaces of hyadic spaces, Proc. Amer. Math. Soc. 104 (1988), 635-640. Zbl0691.54013
  6. [6] J. Gerlits, On subspaces of dyadic compacta, Studia Sci. Math. Hungar. 11 (1976), 115-120. Zbl0433.54003
  7. [7] J. Gerlits, On a generalization of dyadicity, ibid. 13 (1978), 1-17. Zbl0475.54012
  8. [8] I. Juhász, Cardinal Functions in Topology--Ten Years Later, Math. Centre Tracts 123, Mathematisch Centrum, Amsterdam, 1980. Zbl0479.54001
  9. [9] I. Juhász and S. Shelah, π ( X ) = δ ( X ) for compact X , Topology Appl. 32 (1989), 289-294. 
  10. [10] W. Kulpa and M. Turzański, Bijections onto compact spaces, Acta Univ. Carolin. Math. Phys. 29 (1988), 43-49. Zbl0676.54028
  11. [11] G. Plebanek, Compact spaces that result from adequate families of sets, Topology Appl. 65 (1995), 257-270. Zbl0869.54003
  12. [12] G. Plebanek, Erratum to 'Compact spaces that result from adequate families of sets', ibid. 72 (1996), 99. 
  13. [13] B. Shapirovskiĭ, Maps onto Tikhonov cubes, Russian Math. Surveys 35 (1980), no. 3, 145-156. Zbl0462.54013
  14. [14] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 110 (1979), 407-438. Zbl0393.46019
  15. [15] S. Todorčević, Remarks on cellularity in products, Compositio Math. 57 (1986), 357-372. Zbl0616.54002
  16. [16] M. Turzański, On generalizations of dyadic spaces, Acta Univ. Carolin. Math. Phys. 30 (1989), 153-159. Zbl0713.54040
  17. [17] M. Turzański, Cantor cubes: chain conditions, Prace Nauk. Uniw. Śląsk. Katowic. 1612 (1996). Zbl0862.54021

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