I’m wondering if the set of non-negative sequences which sum to 1 is compact under the product (or weak) topologies.
That is: $(a_1,a_2,...)$ such that $\sum_n a_n=1$ where $a_n \geq 0 \forall n$
I realize that this set of sequences is not closed under the $l^1$-topology because the following sequence of sequences has no convergent subsequence.
(1,0,0,0,0,0,0,...) (0,1,0,0,0,0,0,...) (0,0,1,0,0,0,0,...)
But, this kind of example does not apply to the product topology, which is not metrizable.