A separable metric space is called cohesive if every point of the space has a neighbourhood that fails to contain nonempty clopen subsets of the space. A topological group is cohesive if and only if it is not zero-dimensional. We present a homogeneous, one-dimensional, almost zero-dimensional space that is not cohesive.
We also show that a complete homogeneous space is cohesive if and only if it is not zero-dimensional.
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