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From: Julien Santini on 1 Mar 2005 12:03
> I would recommend Hatcher's Algebraic Topology, which freely available
over
> the web (http://www.math.cornell.edu/~hatcher/AT/ATpage.html), instead.

This is the textbook our professor (from Cambridge, by the way) is taking
his material from. I was kind of scared by the length of the book, but I'm
definitely going to stick to it now.

--
Julien Santini


From: Stuart M Newberger on 1 Mar 2005 17:00

Zbigniew Fiedorowicz wrote:
> 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.

Thank you for showing me how to define the metric on a quotient space
which is new to me. Regards,Stuart M Newberger

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