LowerSemicontinuousAt — Mathlib

English

Existence of a basis with compact intersections implies quasi-separatedness.

Русский

Существование базиса с компактными пересечениями влечет квазиразделимость.

LaTeX

$$basis isTopologicalBasis ∧ ∀ i j, IsCompact (b_i ∩ b_j) ⇒ QuasiSeparatedSpace α$$

Lean4

/-- A real function `f` is lower semicontinuous at `x` if, for any `ε > 0`, for all `x'` close
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space,
using an arbitrary `y < f x` instead of `f x - ε`. -/
def LowerSemicontinuousAt (f : α → β) (x : α) :=
  ∀ y < f x, ∀ᶠ x' in 𝓝 x, y < f x'

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