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.



What is an example of a non-regular, totally path-disconnected Hausdorff space?
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.


