Component is not necessarily Path Component

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $C$ be a component of $T$.

Then it is not necessarily the case that $C$ is also an path component of $T$.

Proof
Let $C$ be the closed topologist's sine curve embedded in the real Euclidean plane.

From Closed Topologist's Sine Curve is Connected, $C$ is connected in $T$

Therefore $C$ is a component in the subspace of $T$ induced by $C$.

From Closed Topologist's Sine Curve is not Path-Connected, $C$ is not path-connected.

Therefore $C$ is not a path component in the subspace of $T$ induced by $C$.

Hence the result.