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  • 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.