Relationship between Component Types

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

Let $A \subseteq X$.

Let:
 * $A$ be the set of arc components of $T$


 * $P$ be the set of path components of $T$


 * $C$ be the set of components of $T$


 * $Q$ be the set of quasicomponents of $T$.

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

That is:


 * Every arc component is a path component


 * Every path component is a component


 * Every component is a quasicomponent.

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 $f \in P$.

So by definition each arc component is also a path component.

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.