JP Journal of Geometry and Topology
Volume 2, Issue 3, Pages 185 - 201
(November 2002)
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RECONCILING QUASI-UNIFORM COMPACTIFICATION WITH SMIRNOV COMPACTIFICATION
Salvador Romaguera (Spain) and Miguel A. Sanchez-Granero (Spain)
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Abstract:
A Hausdorff
*-compactification
of a (Hausdorff) quasi-uniform space (X,
U)
is a compact Hausdorff quasi-uniform space (Y,
V)
that has a T(V*)-dense
subspace quasi-isomorphic to (X,
U),
where V*
denotes the coarsest uniformity finer than V.
We
prove that if (X,
U)
has a Hausdorff *-compactification, then there is a proximity
rU
on X such that the *-compactification
of (X,
U)
is equivalent to the Smirnov
compactification of (X,
rU).
Furthermore, the Smirnov compactification of (X,
rU)
is the greatest Hausdorff (quasi-uniform)
compactification of (X,
U).
We also show that if, in addition, (X,
U)
is transitive, then the Smirnov
compactification of (X,
rU)
is a Wallman type compactification. |
| Keywords and phrases: quasi-uniform, Hausdorff compactification, proximity, the Smirnov compactification, Wallman type compactification. |
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