Comparing the nine topologies on a three-point set

Write the nine topologies from §12 Example 1 as \[ \begin{aligned} \mathcal T_1&=\{\varnothing,X\},\\ \mathcal T_2&=\{\varnothing,\{a\},\{a,b\},X\},\\ \mathcal T_3&=\{\varnothing,\{b\},\{a,b\},\{b,c\},X\},\\ \mathcal T_4&=\{\varnothing,\{b\},X\},\\ \mathcal T_5&=\{\varnothing,\{a\},\{b,c\},X\},\\ \mathcal T_6&=\{\varnothing,\{b\},\{c\},\{a,b\},\{b,c\},X\},\\ \mathcal T_7&=\{\varnothing,\{a,b\},X\},\\ \mathcal T_8&=\{\varnothing,\{a\},\{b\},\{a,b\},X\},\\ \mathcal T_9&=\mathcal P(X). \end{aligned} \] Write \(\mathcal T<\mathcal T'\) when \(\mathcal T\subsetneq\mathcal T'\), that is, when \(\mathcal T'\) is strictly finer than \(\mathcal T\). The strict inclusions are generated transitively by \[ \mathcal T_1<\mathcal T_4<\mathcal T_3<\mathcal T_6<\mathcal T_9, \] \[ \mathcal T_1<\mathcal T_4<\mathcal T_8<\mathcal T_9, \] \[ \mathcal T_1<\mathcal T_7<\mathcal T_2<\mathcal T_8<\mathcal T_9, \] and \[ \mathcal T_1<\mathcal T_5<\mathcal T_9. \] In addition, \(\mathcal T_7<\mathcal T_3\) and hence \(\mathcal T_7<\mathcal T_6\). These inclusions and their transitive consequences are all the comparable pairs.

Every other pair is incomparable. For each such pair \((\mathcal T,\mathcal T')\), the table gives a set in \(\mathcal T\) but not in \(\mathcal T'\), and a set in \(\mathcal T'\) but not in \(\mathcal T\):