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From: Stuart M Newberger on 28 Feb 2005 19:34

Lasse wrote:
> OK, I see that I made a typo: it should have been |i-z| rather than
> |1-z| in the definition of the metric. Whoops. Does this make more
> sense now?
>
> Lasse

A((t,*),z)= t+|z| ,and B((t,*),z)=1-t+|i-z| (with the Euclidean metric
in the other cases) both define metrics on your set.However in general
the minimum of 2 metrics is not a metric (easy example on a 3 point
space) .The maximum of 2 metrics is a metric (but will not give the
convergence property regarding convergence to (1,*) that you need). Why
then does d= Min(A,B) define a metric? Does the triangle inequality
hold? I doubt it but could be wrong for this special case.Regards
,Stuart M Newberger

From: Lasse on 1 Mar 2005 03:09
>
> A((t,*),z)= t+|z| ,and B((t,*),z)=1-t+|i-z| (with the Euclidean
metric
> in the other cases) both define metrics on your set.However in
general
> the minimum of 2 metrics is not a metric (easy example on a 3 point
> space) .The maximum of 2 metrics is a metric (but will not give the
> convergence property regarding convergence to (1,*) that you need).
Why
> then does d= Min(A,B) define a metric? Does the triangle inequality
> hold? I doubt it but could be wrong for this special case.Regards
> ,Stuart M Newberger

In that case, you do not understand the example. In any metric space,
given two points z,w, you can form a quotient topological space by
identifying z and w and define a metric on this space by

dtilde( x,y) = min( d(x,y), d(x,z) + d(w,y), d(x,w) + d(z,y) ).

This function is built so that it will respect the triangle inequality.
If you want to check it, it's somewhat boring and will certainly
involve checking several cases, but is easily doable by hand.

In the example above, I first identified one endpoint of the interval
[0,1] with 0, and then identified the other endpoint with i. A better
phrased description of a similar space was given by Zbigniew
Fiedorowicz above.

Lasse

From: Stuart M Newberger on 1 Mar 2005 06:54

Lasse wrote:
> >
> > A((t,*),z)= t+|z| ,and B((t,*),z)=1-t+|i-z| (with the Euclidean
> metric
> > in the other cases) both define metrics on your set.However in
> general
> > the minimum of 2 metrics is not a metric (easy example on a 3 point
> > space) .The maximum of 2 metrics is a metric (but will not give
the
> > convergence property regarding convergence to (1,*) that you need).
> Why
> > then does d= Min(A,B) define a metric? Does the triangle
inequality
> > hold? I doubt it but could be wrong for this special case.Regards
> > ,Stuart M Newberger
>
> In that case, you do not understand the example. In any metric space,
> given two points z,w, you can form a quotient topological space by
> identifying z and w and define a metric on this space by
>
> dtilde( x,y) = min( d(x,y), d(x,z) + d(w,y), d(x,w) + d(z,y) ).
>
> This function is built so that it will respect the triangle
inequality.
> If you want to check it, it's somewhat boring and will certainly
> involve checking several cases, but is easily doable by hand.
>
> In the example above, I first identified one endpoint of the interval
> [0,1] with 0, and then identified the other endpoint with i. A better
> phrased description of a similar space was given by Zbigniew
> Fiedorowicz above.
>
> Lasse

Thank you,yes I was not familiar with that metric on a quotient space
and I wasnt thinking of your example in terms of a quotieint space ,but
now that its cleared up I see that your example is very nice as is that
of Z.Fiedorowicz. Regards,Stuart M. Newberger

From: Zbigniew Fiedorowicz on 1 Mar 2005 09:27
Lasse wrote:
>>the minimum of 2 metrics is not a metric (easy example on a 3 point
>>space)
>>Why then does d= Min(A,B) define a metric? Does the triangle inequality
>>hold? I doubt it but could be wrong for this special case.Regards
>>,Stuart M Newberger
>
>
> In that case, you do not understand the example. In any metric space,
> given two points z,w, you can form a quotient topological space by
> identifying z and w and define a metric on this space by
>
> dtilde( x,y) = min( d(x,y), d(x,z) + d(w,y), d(x,w) + d(z,y) ).

There is a very standard example of a metric defined by taking
minimum/infimum, the case of a connected Riemannian manifold.
Recall that a Riemannian manifold is a manifold with a Riemannian
metric, which is a metric in a different sense (than that of a
metric space). Namely a Riemannian metric endows tangent vectors
with a length. This allows us to assign a length to smooth curves
in the manifold. We can then endow the manifold with the structure
of a metric space by defining the distance between two points to
be the infimum of the lengths of all smooth paths joining the two
points.

More pertinent to what we are discussing here, is the notion of
a quotient metric. Given an equivalence relation on a metric space,
we can define a metric on the quotient space as follows:

d'([x],[y]) = inf(d(u_0,u_1)+d(u_2,u_3)+...+d(u_{2n},u_{2n+1}))

where the infimum is taken over all finite sequences (u_0,u_1,...,u_{2n+1})
with [x]=[u_0], [y]=[u_{2n+1}], [u_{2i-1}]=[u_{2i}], i=1,2,...,n.
Strictly speaking this generally only defines a pseudometric, i.e.
d'([x],[y])=0 does not necessarily imply that [x]=[y]. Note that
the triangle inequality
d'([x],[y]) <= d'([x],[z]) + d'([z],[y])
can be easily deduced by noting that the right hand side of the
inequality can be obtained by taking the infimum over a smaller
set of sequences (u_0,u_1,...,u_{2n+1}) than the left hand side,
namely those where [u_{2i-1}]=[u_{2i}]=[z] for at least one value of i
between 1 and n.

However for nice enough equivalence relations, d' is a metric
and induces the quotient topology. In the particular case of
an equivalence relation given by identifying two points in the
metric space, d' reduces to dtilde above.
From: Zbigniew Fiedorowicz on 1 Mar 2005 11:23
Julien Santini wrote:
> The theorem I mentioned is theorem 4.8 (Hocking & Young) and the proof is
> "left to the reader" (it is clearly implied it must be more or less
> trivial). However, it seems clear to me that, using this interpretation of
> starlike, a counterexample can be found easily. I really can't tell.

While Hocking and Young have some nice material which is hard to find
in other topology texts (e.g. Hahn-Mazurkiewicz Theorem), I would not
recommend it as a text for learning algebraic topology. I find the exposition
obscure and in places misleading, if not downright incorrect as the statement
above.

I would recommend Hatcher's Algebraic Topology, which freely available over
the web (http://www.math.cornell.edu/~hatcher/AT/ATpage.html), instead.
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