```Date: Jan 16, 2013 10:23 PM
Author: William Elliot
Subject: Re: Lovelock and Rund: Star shaped set of points on a manifold

On Wed, 16 Jan 2013, Hetware wrote:> "for a given point P on X_n let us choose our coordinates such that> x^1=...=x^n=0 at P, after which we construct an open set U on X_n which is> defined by the property that for any point Q element of U with coordinates> x^h, the segment consisting of the points with coordinates tx^h, 0<=t<=1, is> also contained in U."> > I'm having a bit of trouble grasping that concept.  Let's take R^2, for> example.  I choose a point in the middle of my paper, and call it {0,0}.  I> now declare it to be a member of some open set U.  A star set is an open set that's radially convex.A star set with p as it's center is an open set U with p in U and for all x in U, the line from p to x lies within U.> The smallest possible U is some infinitesimal open disk centered on P.  There is no smallest possible star set with p as it centerunless p is an isolated point.  Infinitesimal open disks arean allusion.  Open disks with a (sufficiently) tiny radiusis what to think instead of those allusive disks.
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