Relationship between Component Types

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $p \in S$.

Let:
 * $A$ be the arc component of $p$


 * $P$ be the path component of $p$


 * $C$ be the component of $p$


 * $Q$ be the quasicomponent of $p$.

Then:
 * $A \subseteq P \subseteq C \subseteq Q$

In general, the inclusions do not hold in the other direction.

Proof
Let $f \in A$.

By Arc in Topological Space is Path we have that $f \in P$.

That is, $A \subseteq P$.

Let $f \in P$.

From Path-Connected Space is Connected we have directly that $P \subseteq C$.

Let $f \in C$.

From Connected Space is Connected Between Two Points we have directly that $C \subseteq Q$.

Hence the result.

Also see

 * Path Component is not necessarily Arc Component
 * Component is not necessarily Path Component
 * Quasiomponent is not necessarily Component