gn.general topology - nonhausdorff dimension

What you have defined is what general topologists call an ordinal invariant of a topological space. This is just what it sounds like: an assignment of an ordinal number to every topological space in such a way that homeomorphic spaces get assigned equal ordinals.

Knowing this terminology may help you search the literature to see whether your invariant already appears. For instance, you might look here:

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Kannan, V.
Ordinal invariants in topology.
Mem. Amer. Math. Soc. 32 (1981), no. 245, v+164 pp.

MathReview by S.P. Franklin:

What follows is the text of an advertising blurb for this manuscript used by the AMS. Filtering out the obvious sales pitch leaves a general but accurate account of the contents: ``The concept of the order of a map is so powerful as to form a base for the unification of several ordinal invariants in topology. In this work, the author shows that the derived length of scattered spaces, sequential order of sequential spaces, etc., can all be described in terms of this notion. This view helps to extend them so as to be defined for all topological spaces without missing their most significant properties, to dualize them, to perceive them in the background of category theory and to obtain a lot of new information. In this self-contained work the author incidentally comes across some close connections among such apparently unrelated areas of topology as Čech closures, coreflective subcategories, special morphisms and the ordinal invariants mentioned above. The notion of $E$-order introduced here provides a unification of such invariants as sequential order, $k$-order, $m$-net-order and so on. This theory is not only more satisfactory than the earlier attempts of unification but also encompasses them as subcases.''

The list beginning with sequential order should also include the derived order.

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To me it seems possible that this definition and your result -- that every ordinal number arises in this way from a topological space -- could be publishable, if written up in a succinct and attractive way.