Cantor Connectedness of Uniform Spaces

A uniformly connected space or Cantor connected space is a uniform space such that every uniformly continuous function from to a discrete uniform space is constant. A uniform space is called uniformly disconnected if it is not uniformly connected.
We state an equivalent definition of Cantor connectedness based on definition of uniform space by uniform coverings. We prove several properties of Cantor connectedness using the notion of chain.