Connectedness of Points 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 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
We have that $\set a \subseteq S$ is a (degenerate) connected set of $S$.

So $a$ is in the same connected set as itself and so $a \sim a$.

So $\sim$ has been shown to be reflexive.

Symmetry
If $x \sim y$ then $x$ is in the same connected set as $y$ by definition.

Trivially it follows that $y$ is in the same connected set as $x$ and so $y \sim x$.

So $\sim$ has been shown to be symmetric.

Transitivity
Let $x \sim y$ and $y \sim z$.

Then by definition:
 * $x$ is in the same connected set as $y$


 * $y$ is in the same connected set as $z$

By Union of Connected Sets with Non-Empty Intersections is Connected it follows that $x$ is in the same connected set as $z$.

Thus so $x \sim z$.

So $\sim$ has been shown to be transitive.

$\sim$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.