Copyright © 2012 Lei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a metric space and a sequence of continuous maps that converges uniformly to a map . We investigate the transitive subsets of whether they can be inherited by or not. We give sufficient conditions such that the limit map has a transitive subset. In particular, we show the transitive subsets of that can be inherited by if converges uniformly strongly to .