Suppose X is a space and A1⊆A2⊆A3⊆...⊂X is a sequence of subspaces each of which is closed in X and such that X≅lim→nAn (i.e. U is open in X if and only if U∩An is open in An 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 An is Hausdorff for each n, then must X also be Hausdorff?
The answer is no. H. Herrlich showed, in 1969, that even if you consider each An a completely regular space, the direct limit may fail to be Hausdorff. However if all An are T4 – spaces then X is a T4 – space (it’s not hard to prove this).
A comment about the definition of direct limit. Usually, in category theory, we call direct limit a colimit of a directed family of objects. Using this terminology it’s well known that the category of Hausdorff spaces isn’t closed under direct limits. You can find some examples in Dugundji’s ‘Topology’ (a shame it’s out of print). The definition you are using is very particular, so Herrlich’s example is special.
In this paper D. Hajek and G. Strecker exhibit sufficient conditions for the Hausdorff property to be preserved under direct limits.