We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group F has open kernel. A similar claim holds when F is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group.
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Slender group,Thompson's group,word hyperbolic group,completely metrizable group,locally compact Hausdorff group
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