TITLE: Are almost sequential spaces sequential?
QUESTION [8 upvotes]: A topological space $X$ is called
$\bullet$ sequential if for each non-closed subset $A\subset X$  there exists a sequence $\{a_n\}_{n\in\omega}\subset A$ that converges to a point $a\notin A$;
$\bullet$ almost sequential if each point $x\in X$ is contained in a dense sequential subspace of $X$.

Question. Is there an almost sequential regular space $X$ which is not sequential (and moreover, contains a closed countable subspace $F\subset X$ that has no non-trivial convergent sequences)?

REPLY [4 votes]: $2^\kappa$ for uncountable $\kappa$ is a ZFC example of an almost sequential non-sequential space. 
The easiest way to see why it's not sequential is to note that there is a set $A \subset 2^\kappa$ and a point $p \in \overline{A}$ such that $p$ is not in the closure of any countable subset of $A$. It suffices to take $A=\{x \in 2^\kappa: |x^{-1}(1)| < \aleph_0 \}$ and $p \in 2^\kappa$ to be the function constantly equal to 1.
To see why it's almost sequential, let $x$ be any point in $2^\kappa$ and let $D=\{y \in 2^\kappa: |\{\alpha < \kappa: x(\alpha) \neq y(\alpha) \}| < \aleph_0 \}$. Then $D$ is a sequential (even Fréchet-Urysohn) dense subspace of $2^\kappa$ which contains $x$.