Connectedness of Points is Equivalence Relation

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Theorem

Let $T = \left({S, \tau}\right)$ 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 $\left\{{a}\right\} \subseteq S$ 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.

$\Box$


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.

$\Box$


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.

$\Box$


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

Hence by definition it is an equivalence relation.

$\blacksquare$


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