arcwise connected open set

Prove:

Any connected open subset S of E^n is arcwise connected.

Definition: A set in E^n is said to be connected if it is impossible to split S into two disjoint sets A and B, neither one empty, without having one of the sets contain a boundary point of the other.

Assumption: Any two points in a spherical neighborhood of a point in S can be connected by a straight line.

Let A consist of a point x in S and all other points arcwise connected to x. Let B be all other points of S (S-A).

Consider a spherical neighborhood of a boundary point of A and B (Definition). Let z be in A and w in B. There is a straight line connecting z and w (assumption) because the boundary point and spherical neighborhood are in S. Therefore w in A- contradiction. Therefore B is empty.

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