Abstract
We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.
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Rosendal, C. Automatic continuity in homeomorphism groups of compact 2-manifolds. Isr. J. Math. 166, 349–367 (2008). https://doi.org/10.1007/s11856-008-1034-x
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DOI: https://doi.org/10.1007/s11856-008-1034-x