Uniform connectedness and uniform local connectedness for lattice-valued uniform convergence spaces
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Abstract:
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.
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Journal title
volume 13 issue 3
pages 95- 111
publication date 2016-06-30
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