LowerSemicontinuousOn — Mathlib

English

If s is open, then IsQuasiSeparated s is equivalent to QuasiSeparatedSpace s.

Русский

Если s открыто, то IsQuasiSeparated s эквивалентно QuasiSeparatedSpace s.

LaTeX

$$IsOpen s ⇒ (IsQuasiSeparated s ↔ QuasiSeparatedSpace s)$$

Lean4

/-- A real function `f` is lower semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`,
for all `x'` close enough to `x` in `s`, 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 LowerSemicontinuousOn (f : α → β) (s : Set α) :=
  ∀ x ∈ s, LowerSemicontinuousWithinAt f s x

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