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I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for the sort of space in the question, by a result from

J. Brazas, The topological fundamental group and hoop earring spaces, 2009, arXiv:0910.3685

but the author doesn't supply an explicit example of such a space.

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2 Answers 2

up vote 6 down vote accepted

One of the easiest examples is the rational numbers with the subspace topology of the real line with the K-topology. Total path disconnectedness is not entirely necessary for multiplication of $\pi_{1}(\Sigma X_{+})$ to fail to be continuous. It just makes the path component space of $X$ equal to $X$, greatly simplifying complications.

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Thanks, Jeremy! I knew it would be something relatively simple, but I have little intuition for point-set topology. If we drop total path disconnectedness, what do we have to assume? –  David Roberts May 4 '10 at 22:22
    
Well...you have to know something about the path component space $\pi_{0}^{top}(X)$ (the quotient of $X$ where path components are identified). It is good enough to know that $\pi_{0}^{top}(X)$ is Hausdorff but not completely regular to be sure that $\pi_{1}^{top}(\Sigma X_+)$ is not a topological group. Since $\pi_{0}^{top}$ is essentially surjective this gives a counterexample for every Hausdorff, non-completely regular space. There are much better examples (locally simply connected metric spaces even!) which would probably be more interesting. These will appear in a new version soon. –  Jeremy Brazas May 6 '10 at 12:49
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There are several such examples in Steen and Seebach, Counterexamples in Topology. The first one I saw is number 60, the "relatively prime integer topology", consisting of the set $\mathbb{Z}^+$ of positive integers, with a basis of sets of the form {b + na} where (a,b)=1 and n is an integer.

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Should we make this a rule? Before asking for a counterexample in topology, you should consult the book COUNTEREXAMPLES IN TOPOLOGY. –  Gerald Edgar May 4 '10 at 15:42
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If the book were freely available on the internet, this would seem reasonable. But it isn't, is it? –  Pete L. Clark May 4 '10 at 22:11
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It's not freely available on the internet, to my knowledge, but, being a Dover book, it is available on the internet for very cheap: amazon.com/Counterexamples-Topology-Lynn-Arthur-Steen/dp/… –  Keenan Kidwell May 5 '10 at 0:45
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@PeteL.Clark A searchable database based on Counterexamples is available for free online: austinmohr.com/spacebook. –  Austin Mohr Apr 22 at 21:25
    
@Austin: great! –  Pete L. Clark Apr 24 at 17:15
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