Originally Posted by

**Hartlw** The question is how do you prove the statement about arcwise connectedness.

Taylor, "General Theory of Functions and Integration"

Taylor gives a long and involved proof, most of which can be followed.

From Taylor:

"Theorem: Let S be an open and connected set in R^k. Then, given any two distinct points x,y in S, there exists a polygonal arc lying in S and having x and y as its end points."

"Let A consist of x and all points a in s such that there exists a polygonal arc lying in S with x and a as its end points. We have to prove that y is in A. We shall assume the contrary and deduce a contradiction. Let B=S-A. Evidentally A unequal 0, B unequal 0, S = AUB, AXB = 0."

To be disconnected, AXB must equal zero (just proved) and AXB' = 0 and A'XB = 0. (Prime means the set of all accumulation points). Then

He proves A is open so that AXB'=0 (no accumulation points in A).

That leaves A'XB =0. That's hard. OK, I'll copy the damn thing:

"If we now prove that A'XB ==0, S will be expressed as the union of the separated sets A, B, contrary to the fact that S is connected, and so we shall have completed our proof. Suppose then that A'XB unequal zero, say z eA'XB. Then zeS, and there exists a spherical neighborhood of z lying entirely in S.This neighborhood must contain a point w in A because zeA'. There is then a polygonal arc L in S joining x to w. Let M be the line segment from w to z, (ends included). Now z is not on L because L is a subset of A and zeB."

Now I get stuck. Continuing with Taylor:

"__The set LXM is compact, and hence there is a point y of L at minimum distance from z. We then obtain a polygonal arc joining x to z by proceeding from x along L until we come to y, and then following the straight line segment from y to z__."

The rest then is easy:

"But this means that zeA, which is a contradiction. Hence we must conclude that A'XB=0, and the proof is complete."

comment: Not only is this an excercise in Rosenlicht, it is also an excercise in Buck, "Advanced Calculus," with of course, no answer. Are these people serious?

What I was really looking for is an understandable proof, or a reference to one. If someone could explain the underlined sentence, that would work and be greatly appreciated.