Opened spangled

An open U of the Euclidean Space or a vector Space normalized is known as spangled compared to a point so has for any point X of U , the segment X , i.e. the whole of the positive Barycentre S of the points has and X is contained in U (this condition ensures that has is inevitably in U ). It returns to same to say that U is stable under the action of the Homothétie S of center has and of report/ratio T for T positive lower than 1.

He will be known as spangled (without more precise details) if he is spangled compared to a point at least.

; Examples:

open a convex is spangled compared to each one of its points.
In the plan, the Complémentaire of a half-line is spangled but is not convex; the complementary one to a point is not spangled.

; Note: The property to be spangled is completely independent of that to be open, but it is that it is useful mainly for the open ones.

; Comment: The principal interest of open spangled is their role in the lemma of Poincaré, according to which all differential Forme on open spangled which is closed is exact. The property to be spangled is not invariant by Homéomorphisme, but the open ones spangled are among the simplest examples and most important of contractile spaces.

See too

Open (topology)
Cohomologie of Rham

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