Compatibility of topology and order without algebra

Stream: maths

[Yaël Dillies]

This is a very speculative question. Are there interesting ways for topology and order to interact without algebra being involved? Let me expand.

[Yaël Dillies]

Fact: The upper closure of a closed set is closed.

This is both true for the lower topology on a preorder (#17037), and for lower-bounded sets in ordered commutative topological (additive) groups (soon to be in #16975).

[Yaël Dillies]

Fact: The closure of an upper set is an upper set.

This is true in ordered commutative topological (additive) groups (#16975) without change to the statement between the additive and multiplicative statements.

[Yaël Dillies]

Fact: Ici a is closed.

This is both true for docs#order_closed_topology and the lower topology on a preorder.

[Yaël Dillies]

Is there any interesting pattern here besides the already-existing docs#order_closed_topology and docs#order_topology?

[Kevin Buzzard]

Just to remark that a topological abelian group has a canonical uniform space structure inducing the topology, but I'm not sure this helps