Separation in direct limits of closed inclusions

Suppose $X$ is a space and $A_1\subseteq A_2\subseteq A_3\subseteq ...\subset X$ is a sequence of subspaces each of which is closed in $X$ and such that $X\cong \varinjlim_{n}A_n$ (i.e. $U$ is open in $X$ if and only if $U\cap A_n$ is open in $A_n$ for each $n$). This topology on $X$ has many names (direct limit, inductive limit, weak topology, maybe more) but I can't seem to find much dealing with separation properties in this general setting. Specifically, I am asking:

If $A_n$ is Hausdorff for each $n$, then must $X$ also be Hausdorff?

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The answer is no. H. Herrlich showed, in 1969, that even if you consider each $A_n$ a completely regular space, the direct limit may fail to be Hausdorff. However if all $A_n$ are T$_4$ - spaces then $X$ is a T$_4$ - space (it's not hard to prove this).