ordConnectedComponent — Mathlib

English

For hx ∈ s and hy ∈ s, OrdConnected(s ∩ Icc x y) iff Ioo x y ⊆ s.

Русский

Для hx ∈ s и hy ∈ s, OrdConnected( s ∩ Icc x y ) эквивалентно Ioo x y ⊆ s.

LaTeX

$$$(hx:\\,x \\in s) (hy:\\,y \\in s) \\Rightarrow \\operatorname{OrdConnected}(s \\cap Icc x y) \\iff Ioo x y \\subseteq s$$$

Lean4

/-- Order-connected component of a point `x` in a set `s`. It is defined as the set of `y` such that
`Set.uIcc x y ⊆ s`. Note that it is empty if and only if `x ∉ s`. -/
def ordConnectedComponent (s : Set α) (x : α) : Set α :=
  {y | [[x, y]] ⊆ s}

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