Research proof index

The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.

This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.

This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.

If f is monotone on a dense set D and is right continuous, then f is monotone. We prove under the assumption that α has a top element ⊤ and ⊤ ∈ D, which is a necessary assumption because otherwise it is possible that ⊤ is an isolated point. This theorem should be also true when α satisfies NoTopOrder α.

Apply the optional stopping theorem to get equation 4. Note that T1 Space is needed to make sure that mesh ι n has order topology.

If the set is inf-closed, elements of countablInfClosure can be written as countable intersections of antitone sequences of sets.