Re: Contractible metric space

From: Lasse (lasse_rempe_at_yahoo.de)
Date: 02/27/05 

Date: 27 Feb 2005 11:08:23 -0800

> That's what I had thought, but my problem is that although I can show
h_x:
> [0,1] -> X, h_x(t) = h(x,t) is continuous (using this definition of
> starlike), I cannot show h_t: X->X, h_t(x) = h(x,t) is continuous.
Working
> in R^2 for instance, and considering an arc (*p), with a point x on
this
> arc, and a neighborhood U of x, why couldn't U intersect (no matter
how
> "small" U is) another arc at point y so that for some fixed t, and
for some
> fixed e>0, abs(h_t(y)-h_t(x))>e (i.e h_t is not continuous) ? That's
what
> bothers me.

How about the following (it would seem that this can be done in R^2,
but perhaps it is easier to define on the abstract level):

How about the following: in R^2, take the union of the line segment
from (0,0) to (1,0); the line segments {1/n} x [1,0] for all natural
numbers n, and a half-circle (or any other curve) from (0,0) to (0,1)
lying in the left half plane (i.e., x < 0) apart from these two
endpoints. Then it would seem that this set is "star-like" by your
definition, but it is clearly not contractible.

You could assume that your space is compact and this problem would go
away, but perhaps you don't want to do that.

Hope this helps,

Lasse 

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