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 center
unless p is an isolated point. Infinitesimal open disks are
an allusion. Open disks with a (sufficiently) tiny radius
is what to think instead of those allusive disks.