English
A nonempty set S is preirreducible iff for every x in S, S ⊆ closure {x}. In a regular space this is equivalent to a stronger local property.
Русский
Множество S не пустое является преднепрерывным тогда и только тогда, когда для каждого x∈S выполняется S ⊆ closure {x}.
LaTeX
$$$IsPreirreducible S \\iff \\forall x\\in S, S \\subseteq \\overline{\\{x\\}}.$$$
Lean4
/-- Two continuous maps into a Hausdorff space agree at a point iff they agree in a
neighborhood. -/
theorem ne_iff_eventually_ne [T2Space Y] {x : X} {f g : X → Y} (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
f x ≠ g x ↔ ∀ᶠ x in 𝓝 x, f x ≠ g x := by
constructor <;> intro hfg
· obtain ⟨Uf, Ug, h₁U, h₂U, h₃U, h₄U, h₅U⟩ := t2_separation hfg
rw [Set.disjoint_iff_inter_eq_empty] at h₅U
filter_upwards [inter_mem (hf.preimage_mem_nhds (IsOpen.mem_nhds h₁U h₃U))
(hg.preimage_mem_nhds (IsOpen.mem_nhds h₂U h₄U))]
intro x hx
simp only [Set.mem_inter_iff, Set.mem_preimage] at hx
by_contra H
rw [H] at hx
have : g x ∈ Uf ∩ Ug := hx
simp [h₅U] at this
· obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 hfg
exact h₁t x h₃t
