Entrywise nonnegative matrices: a closed cone with interior

Idea. The conditions are coordinatewise, so everything reduces to the half-line $[0,\infty)\subset\mathbb{R}$.

Step 1: closed, convex cone. Each $\{a_{ij}\ge0\}$ is a closed half-space (preimage of $[0,\infty)$ under a continuous linear functional); an intersection of closed convex cones is a closed convex cone. In more detail: if $A,B\in N$ and $s,t\ge0$ then $sA+tB$ has entries $sa_{ij}+tb_{ij}\ge0$.

Step 2: positive matrices are interior. Let $A$ have $a_{ij}>0$ for all $i,j$ and put $m=\min_{i,j}a_{ij}>0$. If $\|B-A\|_F<m$ then for every entry $|b_{ij}-a_{ij}|\le\|B-A\|_F<m\le a_{ij}$, so $b_{ij}>0$. Hence $B(A,m)\subset N$ and $A\in\operatorname{int}N$.

Step 3: matrices with a zero entry are not interior. Let $A\in N$ with $a_{ij}=0$ for some $(i,j)$. For $\varepsilon>0$, $A-\varepsilon E_{ij}$ has entry $-\varepsilon<0$ at $(i,j)$, so it lies outside $N$, while $\|A-\varepsilon E_{ij}-A\|=\varepsilon$. So $A\notin\operatorname{int}N$.

Conclusion. $\operatorname{int}N=\{A:a_{ij}>0\ \forall i,j\}$.

Remark.
- This is the same picture as for PSD matrices: closed cone, interior = "strict" version, closure of the interior = the cone (here every nonnegative $A$ is the limit of $A+\varepsilon J$, with $J$ the all-ones matrix).
- Positive matrices are exactly the setting of the Perron-Frobenius theorem.