Show that \(\mathbb{R}_\ell\) and the ordered square satisfy the first countability axiom. (This result does not, of course, imply that they are metrizable.)
Solution
For the lower-limit line, the family \[ \mathcal B_x=\{[x,x+1/n):n\in\mathbb Z_+\} \] is a countable neighborhood basis at \(x\). Indeed every basic neighborhood \([x,b)\) contains one of these for sufficiently large \(n\).
Now let \(I_o^2=I\times I\) with dictionary order. We give countable local bases.
If \(0<y<1\), then \[ \left\{\{x\}\times(y-1/n,y+1/n)\cap I^2:n\ge1\right\} \] forms a local basis at \((x,y)\) after discarding the finitely many terms that leave \(I\).
At a lower endpoint \((x,0)\) with \(x>0\), use intervals \[ \big((x-1/n,1),(x,1/n)\big) \] for large \(n\) with \(x-1/n\ge0\). At an upper endpoint \((x,1)\) with \(x<1\), use \[ \big((x,1-1/n),(x+1/n,0)\big). \] Every order interval about the relevant point contains one of these once \(n\) is large enough. At the minimum \((0,0)\) use the countable initial rays ending at \((0,1/n)\), and at the maximum \((1,1)\) use the corresponding final rays beginning at \((1,1-1/n)\).
Thus every point of the ordered square has a countable local basis, so it is first countable.