I am not that familiar with arguments using ordinals so this may be pretty simple. Let $T$ be a topology on a set $X$. For each ordinal $\alpha$ let $T_{\alpha}$ be a topology on $X$ satisfying $T\subseteq T_{\alpha+1}\subsetneq T_{\alpha}$ (notice the proper inclusion) and $T_{\alpha}=\bigcap_{\beta<\alpha}T_{\beta}$ when $\alpha$ is a limit ordinal. Apparently, this should imply that there is some ordinal $\gamma$ such that $T=T_{\gamma}$. Why is this?
Edit: I see there is a problem with the above question. I will try to right this here. Henno is right that I am referring to a certain $f$ that makes a specific topology containing $T$ coarser but so that the new topology still contains $T$.
New question:
We start with a topology $T_0$ on $X$ containing $T$ and have $T\subseteq T_{\alpha+1}=f(T_{\alpha})$ for successor ordinals. We have $T_{\alpha}=\bigcap_{\beta<\alpha}T_{\beta}$ for limit ordinals. We also have that $T_{\alpha+1}=T$ whenever $T_{\alpha+1}=T_{\alpha}$. Apparently there should be a $\gamma$ such that $T=T_{\gamma}$. Why is this?