- #1
jessicaw
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Homework Statement
For any S in X, show that S is path-connected if and only if there exists p in S such that any point x in S can be joined to p by a path.
Homework Equations
A metric space is path connected if any 2 points can be joined by a path in that metric space.
The Attempt at a Solution
If part: Well..if there exists such p, let a joined to p and b joined to p, then a can be joined to b, so arbitary a,b can be joined?
Only if part: Now any 2 points can be joined by a path in that metric space, so let p be a fixed point lying on the path of ab, so p can be joined to a and can be joined to b, so p can be joined to every point?The above is an attempt but there is some errors in the proof, can you check my attempt? Also i think this question is not so easy(i believe the proof should require use of advanced stuff like continuous function, 2 valued, [0,1],...and the like), right?