Let $(X, {cal U})$ be a quasiuniform space, ${cal K}(X)$ be the family of all nonempty compact subsets of $(X, {cal U})$. In this paper, the notion of compact symmetry for $(X, {cal U})$ is introduced, and relationships between the Bourbaki quasiuniformity and the Vietoris topology on ${cal K}(X)$ are examined. Furthermore we establish that for a compactly symmetric quasiuniform space $(X, {cal U})$ the Bourbaki quasiuniformity ${cal U}_*$ on ${cal K}(X)$ is complete if and only if ${cal U}$ is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasiuniform setting. |
Keywords
Bourbaki quasiuniformity, Vietoris topology, small-set symmetric, compactly symmetric, complete.
Math Review Classification
Primary AMS (1991) Subject Classification---54B20, 54E15.
Last Updated
23/3/97
Length
12 pages
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