In order to apply Kakutani's fixed-point theorem successfully on the causality function that projects each past event onto its causing events, I need to demonstrate that timespace is a non-empty, compact and convex Euclidean n-dimensional space.
If the conjecture is acceptable, it would provide support for Aristotle's theorem about the unmoved mover: The causality function would have a fixed point; in other words, there would be at least one event that causes itself.
Under the model conjectured by the Big Bang, the present universe is the boundary of timespace -- which is simply the history of the universe up till present -- and entirely contained in timespace; so timespace is closed; given the expansion of the universe, timespace is also bounded by the present universe, because the future does not exist yet. The largest universe ever is always the present universe. The universe's age-size would be estimated at 13.7 billion years old and approximately 150 billion lightyears across. Assuming the Big Bang model, timespace would be closed and bounded, and therefore compact.
Does this make sense?