Let $H$ be a real Hilbert space. Let $b,c\in H$, $P\subset H$ be a convex cone and a continuous linear mapping $A:H\rightarrow H$. Consider the following sets: $$ B:=\{(Ax, \langle c,x\rangle:x\in P\}, \quad C= B+\{0_{H}\}\times\mathbb{R}_+. $$
Question 1. Whether $B$ is closed in $H\times\mathbb{R}$ implies $C$ is closed in $H\times\mathbb{R}$ and vice versa? If not, could we construct counterexamples?
Question 2. Consider the following conditions:
(a) $b\in \overline{A(P)}$;
(b) There exists $r\in\mathbb{R}$ such that $(b,r)\in \overline{C}$;
(c) There exists $r\in\mathbb{R}$ such that $(b,r)\in \overline{B}$.
It is easy to verify that (c)$\Rightarrow$(b)$\Rightarrow$(a). Could we construct counterexamples to show that the reverse is not true in general?