Recently Poonen [@P1] constructed threefolds (fibred into rational surfaces over a curve of genus at least 1) such that $X(k)=\emptyset$ but $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}\not=\emptyset$. It is known that $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$ coincides with the set of adelic points surviving the descent obstructions defined by torsors of arbitrary linear algebraic groups (as proved in [@D; @Sk2] using [@H2; @Stoll]).

In 1997 Scharaschkin and the second author independently asked the question whether $X(k)=\emptyset$ if and only if $X({\mathbb{A}}_k)^{\operatorname{Br}}=\emptyset$ when $X$ is a smooth and projective curve. They also asked if the embedding of $X(k)$ into $X({\mathbb{A}}_k)^{\operatorname{Br}}$ defines a bijection between the closure of $X(k)$ in $X({\mathbb{A}}_k)$ and the set of connected components of $X({\mathbb{A}}_k)^{\operatorname{Br}}$. Despite some evidence for these conjectures, it may be prudent to consider also their weaker analogues with $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$ in place of $X({\mathbb{A}}_k)^{\operatorname{Br}}$.

In this note we give an elementary construction of a smooth and projective surface $X$ over an arbitrary number field $k$ that is a counterexample to the Hasse principle and has infinite étale Brauer–Manin set (Section \[HP\]). Even simpler is our counterexample to weak approximation (Section \[WA\]). This is a smooth and projective surface $X$ over $k$ with a unique $k$-point and infinite étale Brauer–Manin set $X({\mathbb{A}}_k)^{\operatorname{\acute{e}t},\operatorname{Br}}$; moreover, already the 