# Imposing psendocompact group topologies on Abeliau groups

Fundamenta Mathematicae (1993)

• Volume: 142, Issue: 3, page 221-240
• ISSN: 0016-2736

top

## Abstract

top
The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m\left(\alpha \right)\le {2}^{\alpha }$. We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$\left(\alpha \right)\le {r}_{0}\left(G\right)\le \gamma \le {2}^{\alpha }$, or α > ω and ${\alpha }^{\omega }\le {r}_{0}\left(G\right)\le {2}^{\alpha }$, then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies ${r}_{0}\left(G\right)\ge m\left(\alpha \right)$.  Theorem 5.2(b). If G is divisible Abelian with ${2}^{{r}_{0}\left(G\right)}\le \gamma$, then G admits at most ${2}^{\gamma }$-many pseudocompact group topologies.  Theorem 6.2. Let $\beta ={\alpha }^{\omega }$ or $\beta ={2}^{\alpha }$ with β ≥ α, and let $\beta \le \gamma <\kappa \le {2}^{\beta }$. Then both ${\oplus }_{\gamma }ℚ$ and the free Abelian group on γ-many generators admit exactly ${2}^{\kappa }$-many pseudocompact group topologies of weight κ. Of these, some ${\kappa }^{+}$-many form a chain and some ${2}^{\kappa }$-many form an anti-chain.

## How to cite

top

Comfort, W., and Remus, I.. "Imposing psendocompact group topologies on Abeliau groups." Fundamenta Mathematicae 142.3 (1993): 221-240. <http://eudml.org/doc/211983>.

@article{Comfort1993,
abstract = {The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m(α) ≤ 2^α$. We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$(α)≤ r_0 (G) ≤ γ ≤ 2^α$, or α > ω and $α^ω ≤ r_0(G) ≤ 2^α$, then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_0(G) ≥ m(α)$.  Theorem 5.2(b). If G is divisible Abelian with $2^\{r_\{0\}(G)\} ≤ γ$, then G admits at most $2^γ$-many pseudocompact group topologies.  Theorem 6.2. Let $β = α^ω$ or $β = 2^α$ with β ≥ α, and let $β ≤ γ < κ ≤ 2^β$. Then both $⊕_γℚ$ and the free Abelian group on γ-many generators admit exactly $2^κ$-many pseudocompact group topologies of weight κ. Of these, some $κ^+$-many form a chain and some $2^κ$-many form an anti-chain.},
author = {Comfort, W., Remus, I.},
journal = {Fundamenta Mathematicae},
keywords = {pseudocompact group; $G_δ$-dense subgroup; singular cardinals hypothesis; torsion-free rank; connected topological group; 0-dimensional group; divisible hull; chain; anti-chain; -dense subgroup; pseudocompact group topology},
language = {eng},
number = {3},
pages = {221-240},
title = {Imposing psendocompact group topologies on Abeliau groups},
url = {http://eudml.org/doc/211983},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Comfort, W.
AU - Remus, I.
TI - Imposing psendocompact group topologies on Abeliau groups
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 3
SP - 221
EP - 240
AB - The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m(α) ≤ 2^α$. We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$(α)≤ r_0 (G) ≤ γ ≤ 2^α$, or α > ω and $α^ω ≤ r_0(G) ≤ 2^α$, then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_0(G) ≥ m(α)$.  Theorem 5.2(b). If G is divisible Abelian with $2^{r_{0}(G)} ≤ γ$, then G admits at most $2^γ$-many pseudocompact group topologies.  Theorem 6.2. Let $β = α^ω$ or $β = 2^α$ with β ≥ α, and let $β ≤ γ < κ ≤ 2^β$. Then both $⊕_γℚ$ and the free Abelian group on γ-many generators admit exactly $2^κ$-many pseudocompact group topologies of weight κ. Of these, some $κ^+$-many form a chain and some $2^κ$-many form an anti-chain.
LA - eng
KW - pseudocompact group; $G_δ$-dense subgroup; singular cardinals hypothesis; torsion-free rank; connected topological group; 0-dimensional group; divisible hull; chain; anti-chain; -dense subgroup; pseudocompact group topology
UR - http://eudml.org/doc/211983
ER -

## References

top
1. [Ban] B. Banaschewski, Local connectedness of extension spaces, Canad. J. Math. 8 (1956), 395-398.
2. [Bau] J. E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 10 (1976), 401-439. Zbl0339.04003
3. [BCR] S. Berhanu, W. W. Comfort and J. D. Reid, Counting subgroups and topological group topologies, Pacific J. Math. 116 (1985), 217-241. Zbl0506.22001
4. [CEG] F. S. Cater, P. Erdős and F. Galvin, On the density of λ-box products, General Topology Appl. 9 (1978), 307-312. Zbl0394.54002
5. [C] W. W. Comfort, Topological groups, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam 1984, 1143-1263.
6. [CvM] W. W. Comfort and J. van Mill, Concerning connected, pseudocompact Abelian groups, Topology Appl. 33 (1989), 21-45.
7. [CRe1] W. W. Comfort and D. Remus, Long chains of Hausdorff topological group topologies, J. Pure Appl. Algebra 70 (1991), 53-72.
8. [CRe2] W. W. Comfort and D. Remus, Pseudocompact topological group topologies, Abstracts Amer. Math. Soc. 12 (1991), p. 289 [= abstract #91T-54-25].
9. [CRe3] W. W. Comfort and D. Remus, Pseudocompact topological group topologies on Abelian groups, ibid. 12 (1991), p. 321 [= abstract #91T-22-66].
10. [CRob] W. W. Comfort and L. C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups, Pacific J. Math. 119 (1985), 265-285. Zbl0592.22005
11. [CRos1] W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291. Zbl0138.02905
12. [CRos2] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. Zbl0214.28502
13. [DS] D. N. Dikranjan and D. B. Shakhmatov, Pseudocompact topologizations of groups, Zb. Rad. (Niš) 4 (1990), 83-93. Zbl0705.22002
14. [vD] E. K. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678-682. Zbl0446.54011
15. [D] R. M. Dudley, Continuity of homomorphisms, Duke Math. J. 28 (1961), 587-594. Zbl0103.01702
16. [Fu] L. Fuchs, Infinite Abelian Groups, Vol. I, Pure Appl. Math. 36, Academic Press, New York 1970.
17. [GJ] L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Math. 43, Springer, New York 1976.
18. [Hal] P. R. Halmos, Comment on the real line, Bull. Amer. Math. Soc. 50 (1944), 877-878. Zbl0061.04404
19. [Haw] D. Hawley, Compact group topologies for R, Proc. Amer. Math. Soc. 30 (1971), 566-572. Zbl0209.06001
20. [HI] M. Henriksen and J. R. Isbell, Local connectedness in the Stone-Čech compactification, Illinois J. Math. 1 (1957), 574-582. Zbl0079.38604
21. [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Grundlehren Math. Wiss. 115, Springer, Berlin 1963.
22. [J] T. Jech, Set Theory, Academic Press, New York 1978.
23. [M1] M. Magidor, On the singular cardinals problem I, Israel J. Math. 28 (1977), 1-31. Zbl0364.02040
24. [M2] M. Magidor, On the singular cardinals problem II, Ann. of Math. 106 (1977), 517-547. Zbl0365.02057
25. [M] O. Masaveu, doctoral dissertation, Wesleyan University, in preparation.
26. [T1] M. G. Tkachenko, On pseudocompact topological groups, Interim Report of the Prague Topological Symposium 2/1987 (1987), p. 18, Czechoslovak Acad. Sci., Prague 1987.
27. [T2] M. G. Tkachenko, Countably compact and pseudocompact topologies on free Abelian groups, Soviet Math. (Izv. VUZ) 34 (1990), 79-86. Russian original: Izv. Vyssh. Uchebn. Zaved. Mat. 1990 (5) (336), 68-75. Zbl0714.22001
28. [We] A. Weil, Sur les espaces à structure uniforme et sur la topologie générale, Publ. Math. Univ. Strasbourg, Hermann, Paris 1937. Zbl63.0569.04
29. [Wu] D. E. Wulbert, A characterization of C(X) for locally connected X, Proc. Amer. Math. Soc. 21 (1969), 269-272. Zbl0174.25603

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.