Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Exact Lean statement
lemma Dense.comap_val_nhdsWithin_Iio_neBot {α : Type*} [TopologicalSpace α] [LinearOrder α]
[OrderTopology α] [DenselyOrdered α] {D : Set α} (hD : Dense D) {a b : α} (hab : b < a) :
((𝓝[Set.Iio a] a).comap ((↑) : D → α)).NeBotFormal artifact
Lean source
lemma Dense.comap_val_nhdsWithin_Iio_neBot {α : Type*} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {D : Set α} (hD : Dense D) {a b : α} (hab : b < a) : ((𝓝[Set.Iio a] a).comap ((↑) : D → α)).NeBot := by refine comap_neBot_iff.2 fun t ht => ?_ obtain ⟨c, hc⟩ := (mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset hab).1 ht obtain ⟨d, hd⟩ := hD.inter_open_nonempty (Set.Ioo c a) isOpen_Ioo (Set.nonempty_Ioo.2 hc.1.2) exact ⟨⟨d, hd.2⟩, hc.2 hd.1⟩- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:920-926
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
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.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
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.
Source project: Brownian motion
Person-level attribution pending.
Dense monotone of is Right Continuous
Dense.monotone_of_isRightContinuous
Project documentation
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.
Source project: Brownian motion
Person-level attribution pending.