Arc-Connectedness is Equivalence Relation
Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $a \sim b $ denote the relation:
- $a \sim b \iff a$ is arc-connected to $b$
where $a, b \in S$.
Then $\sim$ is an equivalence relation.
Proof
Checking in turn each of the criteria for equivalence:
Reflexivity
The definition of arc-connected specifically makes provision that:
- $\forall a \in S: a \sim a$
specifically to ensure that $\sim$ be reflexive.
$\Box$
Symmetry
If $x \sim y$ then $x$ is is arc-connected to $y$ by definition.
In the trivial case of $x = y$, $y \sim x$ by reflexivity.
Suppose that $x \ne y$.
We form the mapping $g: \R \to \R$:
- $\map g x = 1 - x$
which is trivially continuous and injective.
By Composite of Continuous Mappings is Continuous, $g \circ f$ is continuous.
By Composite of Injections is Injection, $g \circ f$ is also injective.
Putting it together we see that $g \circ f$ is a continuous injection which maps $0$ to $y$ and $1$ to $x$.
So $y \sim x$ and $\sim$ has been shown to be symmetric.
$\Box$
Transitivity
Follows directly from Joining Arcs makes Another Arc.
$\Box$
$\sim$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
$\blacksquare$
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