Compact finer topologies and a noncompact mixed product of $K$- and lower-limit intervals

Let \(X\) be a set with two topologies \(T_1\) and \(T_2\), and suppose \(T_1\) is finer than \(T_2\). If \((X,T_1)\) is compact and \((X,T_2)\) is Hausdorff, prove that \(T_1=T_2\).

Let \(I=[0,1]\). Let \((I,K)\) be \(I\) with the subspace topology inherited from the \(K\)-topology on \(\mathbb R\), and let \((I,L)\) be \(I\) with the subspace topology inherited from the lower-limit topology. Let \[ X=\prod_{\alpha\in J}Y_\alpha, \] where each \(Y_\alpha\) is either \((I,K)\) or \((I,L)\). Use part 1 to prove that \(X\) is not compact.