Let $X$ be the space
$(\omega+1)\times(\omega_1+1)-\{(\omega,\omega_1)\}$,
using the product topology. Thus, horizontally, it consists
of a bunch of convergent sequences, converging to the
right-most column; vertically, it consists of convergent
transfinite $\omega_1$-sequences, converging to the top
row; and we have specifically removed the upper right
corner point. This is a Hausdorff space.

Let $A=\omega\times\omega_1\subset X$ be the points of these sequences, without their limits. Any countably
infinite subset of $A$ is bounded below a countable ordinal
and has accumulation points in $X$: either it contains
infinitely many points from one coordinate, in which case
it accumulates on the limit supremum in that coordinate, or it
contains points from infinitely many coordinates, which
will converge to a point of the form $(\omega,\alpha)$ at
the right-most column in $X$. The closure of $A$ in $X$ is
all of $X$, but this includes the top row of points of the
form $(n,\omega_1)$, and this set has no accumulation point
in $X$.