Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics
DOI:
https://doi.org/10.4995/agt.2008.1871Keywords:
Unital-semi-quantale, Unital quantale, (fixed-basis) topology, (fixed-basis) bitopology, Order-isomorphism, Categorical (functorial) embedding, RedundancyAbstract
This paper studies various functors between (lattice-valued) topology and (lattice-valued) bitopology, including the expected “doubling” functor Ed : L-Top → L-BiTop and the “cross” functor E× : L-BiTop → L2-Top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings. Given the greater simplicity of lattice-valued topology vis-a-vis lattice-valued bitopology and the fact that the class of L2-Top’s is strictly smaller than the class of L-Top’s encompassing fixed-basis topology, the class of E×’s makes the case that lattice-valued bitopology is categorically redundant. As a special application, traditional bitopology as represented by BiTop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-Top, where 4 is the four element Boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology.
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