Inner Limit is Closed Set

Theorem

Let $\struct {S, \tau}$ be a Hausdorff topological space.

Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $S$.

Then the inner limit $\liminf_n C_n$ is a closed set.

Proof

According to Inner Limit in Hausdorff Space by Set Closures, the inner limit is given by an arbitrary intersection of closed sets which is closed in the topology $\tau$.

$\blacksquare$