Abstract
A topology τ on a group G is complemented if there exists an indiscrete topology τ' on G such that U∩V={0} for suitable neighborhoods of zero U and V in the topologies τ and τ. The authors give a complementation test for an arbitrary topology. Locally compact groups with complemented topologies have been described. A group all of whose continuous homomorphic images are complete is proved to be compact. A family of 2ω topologies that are pairwise complementary to one another is defined for an arbitrary group.
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References
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Zelenyuk, E.G., Protasov, I.V. Complemented Topologies on Abelian Groups. Siberian Mathematical Journal 42, 465–472 (2001). https://doi.org/10.1023/A:1010466924961
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DOI: https://doi.org/10.1023/A:1010466924961