TITLE: What is an example of a non-regular, totally path-disconnected Hausdorff space?
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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.