A continuous exact set
Leonard Gillman · Proceedings of the American Mathematical Society · 1958
A linearly ordered set 21 is called exact2 if it is not similar to any of its proper subsets. It is easily seen that W is exact if and only if the only similarity transformation of 21 into itself is the identity [9, p. 341]. The first example of an exact set was given by Dushnik and Miller in [5]; for related investigations, such as the decomposition of a set into exact subsets, see the bibliography below. The standard construction is to delete suitably chosen elements from a contionuus set (e.g., the reals), with the result that the exact set so obtained has gaps. Cuesta [2] has asked whether there can exist a continuous exact set.3 In this paper, we construct a continuous exact set of power c=2 0 The proof of exactness, however, requires the arithmetical hypothesis that c is a regular cardinal number, that is, that the sum of fewer than c cardinals, each of which is less than c, is itself less than c. (In particular, this will be the case if the continuum hypothesis is true.)