In this paper, by means of the essential derived operator, the classes of topological spaces whose T0 identification spaces are TDD, TF, TY or TL are characterized. This classes are related with the classes of essentially-T1-spaces (R0 spaces), essentially-TD-spaces and essentially-TUD-spaces, already known.
In this way, we introduce several axioms more general than the axioms between T1 and T0 defined by Aull and Thron, all of them weaker than R0.
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