Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sequential conditions and free topological groups

Authors: Edward T. Ordman and Barbara V. Smith-Thomas
Journal: Proc. Amer. Math. Soc. 79 (1980), 319-326
MSC: Primary 54D55; Secondary 22A99, 54G20
MathSciNet review: 565363
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Most of the results in this paper concern relationships between sequential properties of a pointed topological space (X, p) and sequential properties of the Graev free topological group on X. In particular, it is shown that the free group over a sequential $ {k_\omega }$-space is sequential, and that a nondiscrete sequential free group has sequential order equal to $ {\omega _1}$ (the first uncountable ordinal). The free topological group on a space X which includes a convergent sequence contains a closed subspace homeomorphic to $ {S_\omega }$, a previously studied homogeneous, zero-dimensional sequential space. Finally, it is shown that there is no topological group homeomorphic to $ {S_\omega }$.

References [Enhancements On Off] (What's this?)

  • [A-Fr] A. V. Arhangel'skiĭ and S. P. Franklin, Ordinal invariants for topological spaces, Michigan Math. J. 15 (1968), 313-320. MR 0240767 (39:2112)
  • [B] T. K. Boehme, Linear s-spaces, (Proc. Sympos. Convergence Structures, Univ. of Oklahoma, Norman, 1965), Univ. of Oklahoma, Norman, 1965.
  • [D] R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc. 112 (1964), 483-507; Corrections, ibid. 148 (1970), 623-624. MR 0175081 (30:5266)
  • [F-O-T] T. H. Fay, E. T. Ordman and B. V. S. Thomas, The free topological group over the rationals, General Topology and Appl. 10 (1979), 33-47. MR 519712 (80e:22001)
  • [Fr$ _{1}$] S. P. Franklin, Spaces in which sequences suffice. I, Fund. Math. 57 (1965), 107-115. MR 0180954 (31:5184)
  • [Fr$ _{2}$] -, Spaces in which sequences suffice. II, Fund. Math. 61 (1967), 51-56. MR 0222832 (36:5882)
  • [Fr-T] S. P. Franklin and B. V. S. Thomas, A survey of $ {k_\omega }$-spaces, (Topology Proceedings, vol. 2, no. 1, Proc. 1977 Topology Conf., Baton Rouge, La.), Auburn Univ., Auburn, Ala., 1978, pp. 111-124. MR 540599 (80k:54044)
  • [Gr$ _{1}$] M. I. Graev, Free topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945), 3-64; English transl., Amer. Math. Soc. Transl. (1) 8 (1962), 305-364.
  • [Gr$ _{2}$] -, On free products of topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), 343-354. (Russian) MR 0036768 (12:158c)
  • [H-M] David C. Hunt and Sidney A. Morris, Free subgroups of free topological groups, (Proc. Second Internat. Conf. Theory of Groups, Canberra, 1973), Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, 1974, pp. 377-387. MR 0352317 (50:4804)
  • [H-M-T] J. P. L. Hardy, Sidney A. Morris and H. B. Thompson, Applications of the Stone-Čech compactification to free topological groups, Proc. Amer. Math. Soc. 55 (1976), 160-164. MR 0424994 (54:12952)
  • [M-M-O] John Mack, Sidney A. Morris and Edward T. Ordman, Free topological groups and the projective dimension of a local compact abelian group, Proc. Amer. Math. Soc. 40 (1973), 303-308. MR 0320216 (47:8755)
  • [O$ _{1}$] Edward T. Ordman, Free k-groups and free topological groups, General Topology and Appl. 5 (1975), 205-219. MR 0427525 (55:557)
  • [O$ _{2}$] -, Free products of topological groups which are $ {k_\omega }$-spaces, Trans. Amer. Math. Soc. 191 (1974), 61-73. MR 0352320 (50:4807)
  • [R] M. Rajagopalan, Sequential order and spaces $ {S_n}$, Proc. Amer. Math. Soc. 54 (1976), 433-438. MR 0394576 (52:15377)
  • [S] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133-152. MR 0210075 (35:970)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D55, 22A99, 54G20

Retrieve articles in all journals with MSC: 54D55, 22A99, 54G20

Additional Information

Keywords: Sequential space, sequential order, $ {k_\omega }$-space, free topological group, Graev free topological group, sequential coreflection
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society