Pathological Properties
Solenoids are compact metrizable spaces that are connected, but not locally connected or path connected. This is reflected in their pathological behavior with respect to various homology theories, in contrast with the standard properties of homology for simplicial complexes. In Čech homology, one can construct a non-exact long homology sequence using a solenoid. In Steenrod-style homology theories, the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space.
Read more about this topic: Solenoid (mathematics)
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