Uniform Versions of Index for Uniform Spaces with Free Involutions

Abstract: In this paper, uniform versions of index for uniform spaces equipped with free involutions are introduced. 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 Matou$\check{s}$ek in his book on using the Borsuk-Ulam theorem in 2003. Examples of uniform spaces with finite B-index but infinite uniform version of index are given. It is also seen 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 $(X,T)$. In the end, the concept of coloring is carried over to uniform set up and, to a certain extent, connection between uniform versions of coloring and uniform versions of index is also established.