LowerSemicontinuousWithinAt — Mathlib

English

Quasi-separatedness is preserved under a Homeomorph: α ≃_home β induces equivalent quasi-separatedness.

Русский

Квазиразделимость сохраняется при гомоморфизме: α ≃_home β сохраняет эквивалентность квазиразделимости.

LaTeX

$$quasiSeparatedSpace_congr e : QuasiSeparatedSpace α ↔ QuasiSeparatedSpace β$$

Lean4

/-- A real function `f` is lower semicontinuous at `x` within a set `s` if, for any `ε > 0`, 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 LowerSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) :=
  ∀ y < f x, ∀ᶠ x' in 𝓝[s] x, y < f x'

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