  In this paper we correct a gap of Whyburn type topological lemma and
establish two superior limit theorems. As the applications of our Whyburn type
topological theorems, we study the following Monge-Amp\`{e}re equation
\begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=\lambda^N
a(x)f(-u)\,\, &\text{in}\,\, \Omega,\\
u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial \Omega. \end{array}
\right.\nonumber \end{eqnarray} We establish global bifurcation results for the
problem. We find intervals of $\lambda$ for the existence, multiplicity and
nonexistence of strictly convex solutions for this problem.
