In this paper, uniform versions of index for uniform spaces equipped
with free involutions are introduced and studied. They are mainly based on
Bindex 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 BorsukUlam theorem in 2003. Interrelationships
between these uniform versions of index are established. Examples of
uniform spaces with finite Bindex but infinite uniform version of index
are given. It is shown that for a uniform space X with a free involution T,
a dense Tinvariant 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.
