Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set $X$, i.e. $$I=[\emptyset,A_\lambda,X:A_\lambda\subset X,\lambda\in\Lambda]$$

such that $A_\alpha\subset A_\beta$ whenever $\alpha\le\beta$.

under what condition this chain topology will be compact?

Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set $X$, i.e.

$$I=[\{\emptyset,A_\lambda,X:A_\lambda\subset X,\lambda\in\Lambda\}]$$

such that $A_\alpha\subset A_\beta$ whenever $\alpha\le\beta$.

It is easy to show that $I$ qualifies as a topology on $X$.

under what condition this chain topology will be compact?