Exercises
[0M9] Prerequisites:[06N], [0KX]. We verify that what is expressed in [0GQ] also applies to the ”basis”. Let \({{\mathcal B}}\) be a basis for a topology \(𝜏\) on \(X\); consider the descending order between sets (formally \(A ⪯ B \iff A⊇ B\)); with this order \(({{\mathcal B}},⪯)\) is a directed set, whose minimum is \(∅\). Now suppose the topology is Hausdorff. Then taken \(x∈ X\), let \({{\mathcal U}}=\{ A∈{{\mathcal B}}: x∈ A\} \) be the family of elements of the base that contain \(x\): show that \({{\mathcal U}}\) is a directed set. Show that it has minimum if and only if the singleton \(\{ x\} \) is open.
Solution 1