Uniform versions of index for uniform spaces with free involutions

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. Matoušek 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.