When we attempt to define a causality function, that projects any past event unto the collection ("set") of its causing events, in my impression, a reasonable definition of such causality function, will satisfy Kakutani's fixed-point theorem, and therefore project at least one event unto itself; meaning somehow that the event is a fixed point and therefore causes itself.
This result would be very similar to the result described by Aristotle, in his work "Physics" (350 BC), in which he says that there only one unmoved mover, and that the cause for the unmoved mover is the unmoved mover itself. I somehow suspect that Aristotle would have used Kakutani's fixed-point theorem, if it had been available to him at that time.
I would also like to say that Kant's criticism in his "Critique of Pure Reason" on Aristotle's finding, saying that causality is a property of timespace, and therefore does not exist prior to its existence, is not valid, because during Aritotle's limit approach of the initial timespace point from the right, timespace exists at every point. Kant's criticism is only valid for the left limit approach, which is however, never needed for Aristotle's result.
A full verification of why such causality function would satisfy Kakutani's fixed point theorem:here.
Could anybody verify if this makes sense?