Relationship between Component Types
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Theorem
Let $T = \struct {S, \tau}$ 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, $A \subseteq P$.
$\Box$
Let $f \in P$.
From Path-Connected Space is Connected we have directly that $P \subseteq C$.
$\Box$
Let $f \in C$.
From Connected Space is Connected Between Two Points we have directly that $C \subseteq Q$.
$\Box$
Hence the result.
$\blacksquare$
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Also see
- Path Component is not necessarily Arc Component
- Component is not necessarily Path Component
- Quasicomponent is not necessarily Component
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness