  In this paper we show that topological subgroupoids of Lie groupoids, under
special circumstances are Lie subgroupoids. Giving an example, we indicate that
having the same topological dimension is a necessary condition for topological
subgroupoids to be Lie subgroupoids. Also, we provide some conditions for
double subgroupoids to become double Lie subgroupoids. Moreover, we illustrate
that having the same conditions as the Cartan's theorem for Lie groups, helps
us prove the same theorem for generalized Lie groups.

Differential Geometry