Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$ so that $a_{n} \in A$ but $b \notin A$.

Working in the topological category TOP, we assume the group $G$ is a sequential space, inversion is continuous, and group multiplication is continuous with respect to the standard product topology on $G \times G$.

Must $G \times G$ be a sequential space?

A `yes' answer would provide a sharp dividing line between sequential topological groups in TOP, and sequential topological groups in SEQ\TOP.

(SEQ is a category in which the standard product topology of $G \times G$ is refined to ensure $G \times G$ is sequential).

separableFrechet group is metrizable. math.cornell.edu/~justin/Ftp/Malykhin.pdf At the end, they ask the question (credited to Hrusak) of whether this might hold under Todorcevic's Open Coloring Axiom. I'm not sure what the current state of this is. $\endgroup$ – Iian Smythe Oct 17 '13 at 1:07