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Topological semigroups/monoids: topological space endowed with a continuous semigroup/monoid structure, or, equivalently, semigroup/monoid endowed with a compatible topology.

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Spaces with no topological monoid structure which are homotopy equivalent to topological mon...

See here: Algebras over the little disks operad The question asked there was different/more general, but the answers actually address the restricted question asked here.
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