It might be worth going back to when the term Event was first introduced by Derrida in his paper Structure, Sign, and Play in the Discourse of the Human Sciences presented at a conference in the US to investigate & ratify the idea of Structuralism, but was soon recognised as the beginning of post-structuralism primarily due to Derridas paper. He writes:
Perhaps something has occurred in the history of the concept of structure that could be called an "event," if this loaded word did not entail a meaning which it is precisely the function of structural-or structuralist-thought to reduce or to suspect. But let me use the term "event" anyway, employing it with caution and as if in quotation marks. In this sense, this event will have the exterior form of a rupture and a redoubling.
If this is so, the whole history of the concept of structure, before the rupture I spoke of, must be thought of as a series of substitutions of center for center, as a linked chain of determinations of the center. Successively, and in a regulated fashion, the center receives different forms or names. The history of metaphysics, like the history of the West, is the history of these metaphors and metonymies.
The event I called a rupture, the disruption alluded to at the beginning of this paper, would presumably have come about when the structurality of structure had to begin to be thought, that is to say, repeated, and this is why I said that this disruption was repetition in all of the senses of this word.
Now compare this to Badious 'Event - something that disrupts the current situation'.
In the wikipedia entry to Badiou, they write that Badiou tries
to reconcile a notion of the subject with ontology, and in particular post-structuralist and constructivist ontologies. A frequent criticism of post-structuralist work is that it prohibits, through its fixation on semiotics and language, any notion of a subject.
the place of the event – which is seen as a rupture in being – through which the subject finds realization and reconciliation with truth
Badiou theorises about the Event & Being in terms of axiomatic set theory in a way that is distinct from Philosophers and Mathematicians which make it obscure.