  A Banach space is said to have the ball-covering property (abbreviated BCP)
if its unit sphere can be covered by countably many closed, or equivalently,
open balls off the origin. Let $K$ be a locally compact Hausdorff space and $X$
be a Banach space. In this paper, we give a topological characterization of
BCP, that is, the continuous function space $C_0(K)$ has the (uniform) BCP if
and only if $K$ has a countable $\pi$-basis. Moreover, we give the stability
theorem: the vector-valued continuous function space $C_0(K,X)$ has the (strong
or uniform) BCP if and only if $K$ has a countable $\pi$-basis and $X$ has the
(strong or uniform) BCP. We also explore more examples for BCP on
non-commutative spaces of operators $B(X,Y)$. In particular, these results
imply that $B(c_0)$, $B(\ell_1)$ and every subspaces containing finite rank
operators in $B(\ell_p)$ for $1< p<\infty$ all have the BCP, and $B(L_1[0,1])$
fails the BCP. Using those characterizations and results, we show that BCP is
not hereditary for 1-complemented subspaces (even for completely 1-complemented
subspaces in operator space sense) by constructing two different
counterexamples.
