Wednesday, December 5, 2012

Product of two quotient spaces - an frequent and (in)famous mistake

The first edition of Bourbaki's General Topology (chapter I, §9, p. 56) contains the following theorem.

Proposition 3. Soient $E$, $F$ deux espaces topologiques, $R$ une relation d'équivalence dans $E$, $S$ une relation d'équivalence dans $F$. L'application canonique de l'espace produit $(E/R) \times (F/S)$ sur l'espace quotient $(E\times F)/(R\times S)$ est un homéomorphisme.

It is followed by a very convincing proof. However, the theorem is wrong. The subsequent editions give an example where the spaces are not homeomorphic, even when one of the equivalence relation is equality.

I finally understood where the mistake is. It is in the very statement! Indeed, there is a canonical map, say $h$, between those two spaces, but it goes the other way round, namely from $(E\times F)/(R\times S)$ to $(E/R)\times (F/S)$. This map is continuous, as it should be. But Bourbaki, assuming that the natural canonical map goes the other way round, pretended that $h^{-1}$ is continuous, and embarked in proving that its reciprocal bijection, $h$, is also continuous, what it is...

There are cases where one would like this theorem to holds, for example when one discusses topologies on the fundamental group. Indeed, the fundamental group of a pointed space $(X,x)$ is a quotient of the space of loops based at $x$ on $X$ for the pointed-homotopy relation, hence can be endowed with the quotient of the topology of compact convergence (roughly, uniform convergence on compact sets). Multiplication of loops is continuous. However, the resulting group law on $\pi_1(X,x)$ need not be.

The mistake appears in the recent litterature, see for example this paper, or that one (which has been even featured as «best AMM paper of the year» in 2000...). MathScinet is not aware of the flaws in those papers... Fortunately, MathOverflow is!

5 comments :

  1. The two links for "this paper" and "that one" point to the same...

    There is a procedure for pointing out problems in Math Reviews, probably this should be done for these papers...

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    1. Dear Emmanuel. Thank you for pointing that the second link was incorrect. I did not find any information about this procedure, so I sent an email to the support-service of Math Reviews.

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  2. Fabel's paper http://arxiv.org/abs/0909.3086 provides a very clear counterexample.

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    1. You're perfectly right! Thank you! I knew this paper of Fabel, so I should have told the rest of the story in my post.
      I'll complete it soon. Let's just say that the counterexample is—of course—given by the hawaiian earring.

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    2. This difficulty was one of the reasons for my introducing the category of Hausdorff k-spaces in my 1961 thesis, and discussing the idea of a "category adequate and convenient for all purposes of topology" in my first paper "Ten topologies for X x Y". See the discussion of convenient categories on the ncatlab. Also full details are in my book "Topology and Groupoids" (2006) available from amazon,see my web pages, and the example on quotient spaces was in the 1968 edition of the book.

      Ronnie Brown

      http://www.bangor.ac.uk/r.brown

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