This question arises from my exploratory research on causality.
Let $M=\{ A\in\mathbb R^{n\times n} : \forall i,j~A_{i,j}\geq 0,\sum_{i,j} A_{i,j} = 1 \}$ be the simplex of matrices of size $n\times n$. We equip this set with the weakest topology that makes any linear $f:M\to\mathbb R$ continuous. I am wondering about if the following is true or not.
Is the set of rank $1$ matrices in $M$ a closed set ?
I'm not sure on how to start, any idea is welcome. I cannot use the response in this post, so it feels like if it is closed, the fact that matrices sum to $1$ is relevant.
I would also be very interested in knowing if the set of rank $1$ matrices of $M$ is measurable in the Borel $\sigma$-algebra generated by the above topology, and also if any answer to those would change if the topology was the one induced by a matrix norm.
I think it is closed.
Let $N$ be the set of non-negative vectors in $\mathbb{R}^n$ that sum to $1$. Note that $N$ is compact. Then $$M=\{uv^t|u,v\in N\}$$ Let $e$ be the vector in $\mathbb{R}^n$ with all $1$s. If $\{A_i\in M\}$ is Cauchy, then so is $\{A_i e\}$ and $\{e^tA_i\}$, so their component $u_i$ and $v_i$ have limits in $N$.