Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Minimal convergence spaces


Authors: D. C. Kent and G. D. Richardson
Journal: Trans. Amer. Math. Soc. 160 (1971), 487-499
MSC: Primary 54.22
DOI: https://doi.org/10.1090/S0002-9947-1971-0286063-1
MathSciNet review: 0286063
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We are primarily concerned with minimal $ {\text{P}}$ convergence spaces, where $ {\text{P}}$ is one of the following convergence space properties: Hausdorff, $ {{\text{T}}_2}, \lambda $-regular, $ \lambda $-Urysohn, and first countable, $ \lambda $ an infinite cardinal number. Our conclusions usually resemble the corresponding topological results, but with some deviations ; for instance, a minimal Hausdorff convergence space is always compact, whereas a countable minimal regular convergence space need not be compact.


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

  • [1] M. P. Berri, Minimal topological spaces, Trans. Amer. Math. Soc. 108 (1963), 97-105. MR 27 #711. MR 0150724 (27:711)
  • [2] M. P. Berri and R. H. Sorgenfrey, Minimal regular spaces, Proc. Amer. Math. Soc. 14 (1963), 454-458. MR 27 #2949. MR 0152978 (27:2949)
  • [3] H. J. Biesterfeldt, Jr., Regular convergence spaces, Nederl. Akad. Wentensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 605-607. MR 34 #5048. MR 0205215 (34:5048)
  • [4] C. H. Cook and H. R. Fischer, Regular convergence spaces, Math. Ann. 174 (1967), 1-7. MR 37 #5837. MR 0230274 (37:5837)
  • [5] H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269-303. MR 22 #225. MR 0109339 (22:225)
  • [6] B. V. Hearsey, On extending regularity and other topological properties to convergence spaces, Ph.D. Dissertation, Washington State University, Pullman, Washington, 1968.
  • [7] H. Herrlich, $ {T_v}$-Abgeschlossenheit und $ {T_v}$-Minimalität, Math. Z. 88 (1965), 285-294. MR 32 #1664. MR 0184191 (32:1664)
  • [8] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [9] D. C. Kent, Convergence quotient maps, Fund. Math. 65 (1969), 197-205. MR 40 #3497. MR 0250258 (40:3497)
  • [10] -, Decisive convergence spaces, Fréchet spaces and sequential spaces, Rocky Mt. J. Math. 1 (1971), 367-374. MR 0276916 (43:2656)
  • [11] G. D. Richardson, A Stone-Čech compactification for limit spaces, Proc. Amer. Math. Soc. 25 (1970), 403-104. MR 0256336 (41:992)
  • [12] C. T. Scarborough, Minimal Urysohn spaces, Pacific J. Math. 27 (1968), 611-617. MR 38 #6530. MR 0238254 (38:6530)
  • [13] C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131-147. MR 34 #3528. MR 0203679 (34:3528)
  • [14] R. M. Stephenson, Jr., A countable minimal Urysohn space is compact, Proc. Amer. Math. Soc. 22 (1969), 625-626. MR 39 #6255. MR 0244942 (39:6255)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54.22

Retrieve articles in all journals with MSC: 54.22


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0286063-1
Keywords: Convergence spaces, minimal $ {\text{P}}$ convergence spaces, $ {\text{P}}$-closed convergence spaces, $ \lambda $-regular convergence spaces, $ \lambda $-Urysohn convergence spaces
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society