  A metric space is said to be strongly rigid if no positive distance is taken
twice by the metric. In 1972, Janos proved that a separable metrizable space
has a strongly rigid metric if and only if it is zero-dimensional. In this
paper, we shall develop this result for the theory of space of metrics. For a
strongly zero-dimensional metrizable space, we prove that the set of all
strongly rigid metrics is dense in the space of metics. Moreover, if the space
is the union of countable compact subspaces, then that set is comeager. As its
consequence, we show that for a strongly zero-dimensional metrizable space, the
set of all metrics possessing no nontrivial (bijective) self-isometry is
comeager in the space of metrics.
