# Component is not necessarily Path Component

Jump to navigation
Jump to search

## 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.

$\blacksquare$

## 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