Structures of time directions

In directed topology, one studies topological spaces in which a certain subset of the continuous paths  is singled out as the (time) directed paths. A variation in the set of directed paths, the d-structure, results in a variation of the directed topology of the underlying space.

• If all paths are allowed, the directed topology is ordinary algebraic topology.
• The other extreme is where only constant paths are allowed.
• Removing a subset from the space is equivalent to only allowing paths in that subset to be constant.

Hence, this framework gives a precise way of exploring both variation of the time structure and of the underlying space. In particular the connections between ordinary and directed algebraic topology.    
The interplay between variations of the structure should be addressed via for instance the ordinary algebraic topology of the spaces of directed paths and in particular relative (co)homology  and homotopy. Moreover, the set of d-structures form a lattice, and methods from lattice theory seem to be fruitful and should be explored.