There is a similar question here: Proving a certain implication in the equivalent formulations of Hausdorff spaces but it does not ask for proof of this exact equivalence that I am asking.
My book has the following equivalences for a Hausdorff topological space.
(1) $X$ is Hausdorff
(2) The diagonal relation set $Δ := \left\{ (x,x) | x \in X \right\}$ is a closed set in $X^2$.
(3) Limits of nets in $X$ are unique
I'm looking for a direct proof of (2) => (3).
For (3) => (2) I have the following:
To show Δ is closed, we show that Δ = $\overline{Δ}$.
$\subset$ is always true.
$\supset$: Let $(a, b) \in \overline{Δ}$. Then there is a net in Δ, {$x_j$} that converges to $(a,b)$. Now since {$x_j$} is a net in Δ, for each index $j$, $x_j = (z_j, z_j)$. Observe that in both coordinates, the net in $X$ is exactly the same. So, for the first coordinate, $z_j$ converges to $a$ and for the second coordinate, $z_j$ converges to $b$. Now, by (3), the limit is unique, therefore $a = b$, thus $(a,b) \in Δ$.
The book I use do (1) => (3) => (2) => (1) so I actually do have a roundabout proof for what I want. But is there a direct way?
