Uniform connectedness and uniform local connectedness for lattice-valued uniform convergence spaces

We apply Preuss' concept of mbbe-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly mbbe-connected if the only uniformly continuous mappings from the space to a space in the class mbbe are the constant mappings. We develop the basic theory for mbbe-connected sets, including the product theorem. Furthermore, we define and study uniform local mbbe-connectedness, generalizing a classical definition from the theory of uniform convergence spaces to the lattice-valued case. In particular it is shown that if the underlying lattice is completely distributive, the quotient space of a uniformly locally mbbe-connected space and products of locally uniformly mbbe-connected spaces are locally uniformly mbbe-connected.

Keywords:

L-topology , L-uniform convergence space , Uniform connectedness , Local connectedness