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2 results

  • Nearness Convergence

  • S. A. Naimpally, Mohan L. Tikoo
  • Journal:
    Canadian Mathematical Bulletin / Volume 33 / Issue 3 / 01 September 1990
    Published online by Cambridge University Press:
    20 November 2018, pp. 268-272
    Print publication:
    01 September 1990
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  • In this paper, a concept of nearness convergence is introduced which contains the proximal convergence of Leader as a special case. It is proved that uniform convergence and this nearness convergence are equivalent on totally bounded uniform nearness spaces. One of the features of this convergence is that it lies between uniform convergence and pointwise convergence, and implies uniform convergence on compacta. Some other weaker notions of nearness convergence which are sufficient to preserve nearness maps are also discussed.

  • Topological properties of the scale of a uniform space

  • Part of
  • G. D. Richardson, E. M. Wolf
  • Journal:
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 33 / Issue 1 / August 1982
    Published online by Cambridge University Press:
    09 April 2009, pp. 54-61
    Print publication:
    August 1982
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  • Let (S. U) be a uniform space. This space can be embedded in a complete, uniform lattice called the scale of (S. U). We prove that the scale is compact if and only if S is finite or U = {S × S}. We prove that this statement remains true if compact is replaced by countably compact, totally bounded. Lindelof, second countable, or separable. In the last section of this paper, we investigate the cardinality of the scale and the retracted scale.