  We study the $k$-center problem in a kinetic setting: given a set of
continuously moving points $P$ in the plane, determine a set of $k$ (moving)
disks that cover $P$ at every time step, such that the disks are as small as
possible at any point in time. Whereas the optimal solution over time may
exhibit discontinuous changes, many practical applications require the solution
to be stable: the disks must move smoothly over time. Existing results on this
problem require the disks to move with a bounded speed, but this model allows
positive results only for $k<3$. Hence, the results are limited and offer
little theoretical insight. Instead, we study the topological stability of
$k$-centers. Topological stability was recently introduced and simply requires
the solution to change continuously, but may do so arbitrarily fast. We prove
upper and lower bounds on the ratio between the radii of an optimal but
unstable solution and the radii of a topologically stable solution -- the
topological stability ratio -- considering various metrics and various
optimization criteria. For $k = 2$ we provide tight bounds, and for small $k >
2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an
algorithm to compute the topological stability ratio in polynomial time for
constant $k$.
