Let $E$ be a complex vector bundle over some projective manifold $X$, and assume it is topologically indecomposable (i.e. does not admit any proper non-zero subbundles). Note that I am not assuming that $E$ is holomorphic at the moment.
The slope of $E$ is defined as $\mu(E) = \deg(E)/\operatorname{rank}(E)$, where we can define $\deg(E) = \int_X c_1 (E) \wedge \omega^{n-1} $, where $n$ is the dimension of $X$ and $\omega$ is its Kähler form.
Then, I learned that $E$ is stable if for any proper non-zero sub-bundle $F$ of $E$, we have $\mu (F) < \mu (E)$, and semistable if $\mu (F) \leq \mu (E)$.
Does this definition hold vacuously true in the case of $E$ being topologically indecomposable? In other words, are all topologically indecomposable complex vector bundles stable?