  A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous
map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is
called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection
$\bar U\cap\bar V$ is infinite. We prove that every second-countable Brown
Hausdorff space $X$ admits a stronger topology $\tau'$ such that $X'=(X,\tau')$
is a strongly rigid anticompact Brown space.This construction yields an example
of a countable anticompact Hausdorff space $X$ which is strongly rigid, which
answers two problems posed at MathOverflow. By the same method we construct a
strongly rigid $k_2$-metrizable semi-Hausdorff space containing a non-closed
compact subset, which answers two other problem posed at MathOverflow.
