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In this paper, uniform versions of index for uniform spaces equipped
with free involutions are introduced and studied. They are mainly based on
B-index defined and studied by C.-T. Yang in 1955, index studied by Conner
and Floyd in 1960 and further development well collected by J. Matousek in
his book on using the Borsuk-Ulam theorem in 2003. Interrelationships
between these uniform versions of index are established. Examples of
uniform spaces with finite B-index but infinite uniform version of index
are given. It is shown that for a uniform space X with a free involution T,
a dense T-invariant subspace is capable of determining the uniform version
of index of (X,T). Connections between uniform versions of coloring and
uniform versions of index is also indicated.
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