  Let $A\in \mathcal{B}(X)$ and $B\in \mathcal{B}(Y)$, where $X$ and $Y$ are
Banach spaces, and let $M_{C}$ be an operator acting on $X\oplus Y$ given by
$M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}$. We investigate the limit
point set of the Browder spectrum of $M_{C}$. It is shown that
  $$ acc \sigma_{b}(M_C)\cup W_{acc \sigma_{b}}= acc \sigma_{b}(A)\cup acc
\sigma_{b}(B)$$ where $W_{acc \sigma_{b}}$ is a subsets of $
acc\sigma_{*}(B)\cap acc\sigma_{*}(A)$ and a union of certain holes in $ acc
\sigma_{b}(M_C)$. Furthermore, several sufficient conditions for
$acc\sigma_{b}(M_C)=acc\sigma_{b}(A)\cup acc\sigma_{b}(B)$ holds for every
$C\in \mathcal{B}(Y,X)$ are given.
