Does Zeeman's Fine Topology Exist?
We work on the family of topologies for the Minkowski manifold M. We partially order this family by inclusion to form the lattice Σ(M), and focus on the sublattice Z of topologies that induce the Euclidean metric space on every time axis and every space axis. We analyze the bounds of Z in the lattice Σ(M), in search for its supremum. Our conclusion --that such a supremum does not belong in Z-- is compared with constructive proofs of existence of the fine topology, defined as the maximum of Z and conceived to play an essential role in contemporary physical theories. Essential mathematical and physical questions arise.
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