Tim Maudlin reformulates topology using open lines as the basis rather than open sets.

He thinks this sheds some light on the “arrow of time” question (why does time only move forward, when physical equations can be used to postdict [forensically a bullet, for instance] motion just as easily as to predict [planning a rocket, for instance] motion?).

Notes:

If you don’t like some mathematics, just go make your own.

It’s nice to pay homage to the accepted standards, if’n you’re trying to impress people.

A “topological line” is just a total (linear) order. If you want to join two topological-lines and get another line, you have to make sure you don’t form a circle (two endpoints equate) or a loop-dee-loop. But segments can overlap if they share the same linear order.

A nifty slide on standard topology (as well as Maudlin’s new idea):

His “open lines” basis leads to an interesting conception of neighbourhood on a discrete lattice (shown on a square lattice):