Consider the lower limit topology on \(\mathbb{R}\) and the topology given by the basis \(\mathcal{C}\) of Exercise 8 of §13. Determine the closures of the intervals \(A = (0, \sqrt{2})\) and \(B = (\sqrt{2}, 3)\) in these two topologies.
Solution
Let \(\mathcal T_{\mathcal C}\) be the topology with basis \(\mathcal C=\{[a,b):a<b,\ a,b\in\mathbb Q\}\) of E-NKQY9. Both \(\mathbb R_\ell\) and \(\mathcal T_{\mathcal C}\) are finer than the standard topology, so each closure lies in the standard closure: \(\overline A\subseteq[0,\sqrt2]\) and \(\overline B\subseteq[\sqrt2,3]\). A point \(x\) lies in the closure of a set \(S\) if and only if every basic open set containing \(x\) meets \(S\).
1. In \(\mathbb R_\ell\), \(\overline A=\boxed{[0,\sqrt2)}\).
Every basic set \([c,d)\) containing \(0\) meets \(A\) in \((0,\min(d,\sqrt2))\ne\varnothing\), so \(0\in\overline A\); the points of \((0,\sqrt2)\) lie in \(A\). The basic set \([\sqrt2,2)\) contains \(\sqrt2\) and misses \(A\), so \(\sqrt2\notin\overline A\).
2. In \(\mathbb R_\ell\), \(\overline B=\boxed{[\sqrt2,3)}\).
Every basic set \([c,d)\) containing \(\sqrt2\) has \(c\le\sqrt2<d\) and meets \(B\) in \((\sqrt2,\min(d,3))\ne\varnothing\), so \(\sqrt2\in\overline B\). The basic set \([3,4)\) contains \(3\) and misses \(B\), so \(3\notin\overline B\).
3. A basic set \([a,b)\in\mathcal C\) containing an irrational number \(z\) satisfies \(a<z<b\).
\(a\le z<b\) and \(a\ne z\) because \(a\) is rational.
4. In \(\mathcal T_{\mathcal C}\), \(\overline A=\boxed{[0,\sqrt2]}\).
Every \([a,b)\in\mathcal C\) containing \(0\) meets \(A\) in \((0,\min(b,\sqrt2))\ne\varnothing\), so \(0\in\overline A\). By step 3, every \([a,b)\in\mathcal C\) containing \(\sqrt2\) has \(a<\sqrt2\) and so meets \(A\) in \((\max(0,a),\sqrt2)\ne\varnothing\); hence \(\sqrt2\in\overline A\).
5. In \(\mathcal T_{\mathcal C}\), \(\overline B=\boxed{[\sqrt2,3)}\).
Every \([a,b)\in\mathcal C\) containing \(\sqrt2\) has \(\sqrt2<b\) and so meets \(B\) in \((\sqrt2,\min(b,3))\ne\varnothing\); hence \(\sqrt2\in\overline B\). The set \([3,4)\in\mathcal C\) contains \(3\) and misses \(B\), so \(3\notin\overline B\).
6. Q.E.D.
Steps 1 and 2 give the closures in \(\mathbb R_\ell\), and steps 4 and 5 give the closures in \(\mathcal T_{\mathcal C}\).