What is the question in all of that?
Taylor proves the following (with one incomprehensible step), and Rosenlicht gives it as a problem, a problem!! Is the answer given? Do bears poop in the woods?
Any connected open subset of E^n is arcwise connected.
Definitions from Rosenlicht:
"A metric space E is connected if the only subsets of E which are both open and closed are E and 0. A subset S of a metric space is a connected subset if the subspace S is connected."
"A metric space E is said to be arcwise connected if, given any p,q eE, there is a continuous function f:[0,1] -> E such that f(0)=p , f(1)=q."
Rudin doesn't touch this.
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.
This ought to be a fairly straightforward result, quite reasonable to leave as a problem in a graduate text, and Taylor's proof seems unnecessarily complicated. (I find that surprising, because Taylor is usually an excellent author. His Introduction to functional analysis is still one of the best, 50+ years after its first publication.)
Start as Taylor does, by fixing a point , and defining A to be the set of points in S that can be reached by a path from x. Let B be the complement of A in S. Then A is open. Reason: suppose that ; since S is open, there is a ball B(y,e) centred at y and contained in S. Then every point in that ball belongs to A, because you can connect it to y by a straight line which lies within the ball, and then from y there is a path to x.
A very similar argument shows that B is open: suppose that ; there is a ball centred at z and contained in S. If that ball contains a point then there is a path in S consisting of a straight line from z to w and then from w to x. So – contradiction. Therefore the ball contains no points in A and hence lies entirely in B, showing that B is open.
Thus S is the disjoint union of the open sets A and B. But if S is connected then one of those sets must be empty. It can't be A, because . Therefore B is empty, which means that S is arcwise-connected.
The basic idea in all of this is that the composition of two paths is a path. If there is a path from p to q, and a path from q to r, then you can stitch them together to get a path from p to r.
Drexel28: X is intersection and e is epsilon (belongs to).
Opalg: Thanks so much for reply and very impeccable and transparent proof. The problem with clarity is that it enables someone to see a possible fault, a problem that dosn't occur with Taylor's proof.
If B is the complement of A in S, then B must contain accumulation points of A. In that case, a ball in B centered on an accumulation point can contain points of A without a contradiction.
The point is that B cannot contain accumulation points of A, because that would give a contradiction. The sets A and B are both open and closed in S. Therefore either A and B are completely separate, with no accumulation points in common (which would mean that S is not connected) or B is empty (and therefore S is arcwise-connected).
From Buck, Advanced Calculus: "Definition 4. 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."
So A and B cannot both be open and the argument for B being open fails because a ball exists in B (the boundary point) which can contain a point of A without a contradiction.
But you proved A was open, so its boundary points have to be in B.
From Wolfram Math World:
A connected set is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.
I suspect that's why Taylor is so obstruse: You have to get past the boundary points.
Taylor helps us out here:
Two distinct points in R^k determine a unique straight line- the line joining these two points. If x1 and x2 are the points, the parametric equation of the line is: x=(1-t)x1+x2, t in [0,1]. So if x is any point in S and w is a point in its spherical neighborhood, there is a line from x to w and A is not empty.