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The product of connected spaces is connected, so the product of the two connected components lies in the component of $(x,y)$. Conversely, continuous projections carry that component to connected sets in the factors, which must lie in $C_x$ and $C_y$. These two inclusions prove equality in arbitrary topological spaces.

The connected component of the identity in a Lie group is an open normal subgroup. Connected coordinate neighbourhoods prove openness, and conjugation preserves the component of the identity. Every open identity neighbourhood generates it: the generated subgroup is open, its other cosets are open, and connectedness forces it to be the whole component.

The multiplicity of $a$ in a real tree $T$ is the number of connected components of $T\setminus \{a\}$. A point of multiplicity one is a leaf, one of multiplicity two lies in the interior of an unbranched arc, and one of multiplicity at least three is a branch point.