Title
The Scale of a Quasiuniform Space,Used
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We continue our investigations of the scale of a quasiuniform space, which we had started in an earlier article. We distinguish between the leftsided scale and the twosided scale of a quasiuniform space. While the behavior of the twosided scale of a quasiuniform space X shows similarities with the usual hyperspace of X equipped with its Hausdorff quasiuniformity, the lefthanded scale generalizes the quasiuniform multifunction space of X into itself.For instance the twosided scale of any totally bounded quasiuniform space X is totally bounded, while total boundedness of the leftsided scale of a quasiuniform space X implies that X is finite or indiscrete. Either construction of the scale is based on the idea of the prefilter space of a quasiuniform space. Prefilter spaces of quasiuniform spaces are shown to be bicomplete. It follows that both the leftsided and the twosided scale of a quasiuniform space are bicomplete. Indeed these scales can be used to construct the bicompletion of the T0reflection of the Hausdorff quasiuniformity of a quasiuniform space.
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