  In this paper, by introducing some kind of small loop transfer spaces at a
point, we study the behavior of topologized fundamental groups with the
compact-open topology and the whisker topology, $\pi_{1}^{qtop}(X,x_{0})$ and
$\pi_{1}^{wh}(X,x_{0})$, respectively. In particular, we give necessary or
sufficient conditions for coincidence and being topological group of these two
topologized fundamental groups. Finally, we give some examples to show that the
reverse of some of these implications do not hold, in general.
