  Let $P$ be a simple polygon, then the art gallery problem is looking for a
minimum set of points (guards) that can see every point in $P$. We say two
points $a,b\in P$ can see each other if the line segment $seg(a,b)$ is
contained in $P$. We denote by $V(P)$ the family of all minimum guard
placements. The Hausdorff distance makes $V(P)$ a metric space and thus a
topological space. We show homotopy-universality, that is for every
semi-algebraic set $S$ there is a polygon $P$ such that $V(P)$ is homotopy
equivalent to $S$.
  Furthermore, for various concrete topological spaces $T$, we describe
instances $I$ of the art gallery problem such that $V(I)$ is homeomorphic to
$T$.
