# Every Lusin set is undetermined in the point-open game

Fundamenta Mathematicae (1994)

- Volume: 144, Issue: 1, page 43-54
- ISSN: 0016-2736

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topRecław, Ireneusz. "Every Lusin set is undetermined in the point-open game." Fundamenta Mathematicae 144.1 (1994): 43-54. <http://eudml.org/doc/212014>.

@article{Recław1994,

abstract = {We show that some classes of small sets are topological versions of some combinatorial properties. We also give a characterization of spaces for which White has a winning strategy in the point-open game. We show that every Lusin set is undetermined, which solves a problem of Galvin.},

author = {Recław, Ireneusz},

journal = {Fundamenta Mathematicae},

keywords = {point-open games; Lusin set; additivity of measure; γ-set; gamma set; Hurewitz property; Menger property; -sets; classes of small sets; point-open game},

language = {eng},

number = {1},

pages = {43-54},

title = {Every Lusin set is undetermined in the point-open game},

url = {http://eudml.org/doc/212014},

volume = {144},

year = {1994},

}

TY - JOUR

AU - Recław, Ireneusz

TI - Every Lusin set is undetermined in the point-open game

JO - Fundamenta Mathematicae

PY - 1994

VL - 144

IS - 1

SP - 43

EP - 54

AB - We show that some classes of small sets are topological versions of some combinatorial properties. We also give a characterization of spaces for which White has a winning strategy in the point-open game. We show that every Lusin set is undetermined, which solves a problem of Galvin.

LA - eng

KW - point-open games; Lusin set; additivity of measure; γ-set; gamma set; Hurewitz property; Menger property; -sets; classes of small sets; point-open game

UR - http://eudml.org/doc/212014

ER -

## References

top- [B1] T. Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc. 281 (1984), 209-213. Zbl0538.03042
- [B2] T. Bartoszyński, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), 225-239. Zbl0635.04001
- [BBM] R. H. Bing, W. W. Bledsoe and R. D. Mauldin, Sets generated by rectangles, Pacific J. Math. 51 (1974), 27-36. Zbl0261.04001
- [BRR] L. Bukovský, I. Recław and M. Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology Appl. 41 (1991), 25-40. Zbl0768.54025
- [D] E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam, 1984, 111-167.
- [F] D. H. Fremlin, Cichoń's diagram in: Sém. Initiation à l'Analyse, G. Choquet, M. Rogalski, J. Saint-Raymond, Université Pierre et Marie Curie, Paris, 1983//84, no. 5, 13 pp.
- [FJ] D. H. Fremlin and J. Jasiński, ${G}_{\delta}$-covers and large thin sets of reals, Proc. London Math. Soc. (3) 53 (1986), 518-538. Zbl0591.54028
- [FM] D. H. Fremlin and A. W. Miller, On some properties of Hurewicz, Menger, and Rothberger, Fund. Math. 129 (1988), 17-33. Zbl0665.54026
- [G] F. Galvin, Indeterminacy of point-open games, Bull. Acad. Polon. Sci. 26 (1978), 445-449. Zbl0392.90101
- [GM] F. Galvin and A. W. Miller, γ-sets and other singular sets of real numbers, Topology Appl. 17 (1984), 145-155.
- [M1] A. W. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam, 1984, 201-233.
- [M2] A. W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114; Corrections and additions, ibid. 271 (1982), 347-348.
- [M3] A. W. Miller, Additivity of measure implies dominating reals, Proc. Amer. Math. Soc. 91 (1984), 111-117. Zbl0586.03042
- [M4] A. W. Miller, The Baire category theorem and cardinals of countable cofinality, J. Symbolic Logic 47 (1982), 275-287. Zbl0487.03026
- [M5] A. W. Miller, A characterization of the least cardinal for which Baire category theorem fails, Proc. Amer. Math. Soc. 86 (1982), 498-502. Zbl0506.03012
- [PR] J. Pawlikowski and I. Recław, On parametrized Cichoń's diagram, in preparation. Zbl0847.04004
- [R] I. Recław, On small sets in the sense of measure and category, Fund. Math. 133 (1989), 255-260. Zbl0707.28001

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