Set of Positive Sequences that Sum to 1 is Compact under Product Topology?

Question

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.

Answer 1

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,...)

This sequence is rather a good start. You might try to show that this countable set does not have a limit point, and thus this space is not limit point compact.

But, this kind of example does not apply to the product topology, which is not metrizable.

In fact, product of countably many metric space is metrizable. For example: Show that the countable product of metric spaces is metrizable or Show this metric generates the product topology on $X$. So you can safely apply results which you are known to be true for metrizable spaces.