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Symmetry conditions in terms of open sets. (English) Zbl 0766.54025
Summary: A quasi-uniform space is Smyth symmetric provided that its quasi- proximity is a proximity. This paper shows that a space is Smyth symmetric if and only if it is small-set symmetric and either open symmetric or semi-symmetric in the sense of J. Deák. Every Smyth symmetric quasi-uniformity is quiet, and in the class of totally bounded quasi-uniform spaces both Smyth symmetry and quietness are equivalent to being a uniformity. In the class of small-set symmetric spaces the concepts of open symmetry, semi-symmetry and Smyth symmetry are equivalent.

MSC:
54E05 Proximity structures and generalizations
54E15 Uniform structures and generalizations
54G15 Pathological topological spaces
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