  The percolation threshold and wrapping probability $R_{\infty}$ for the
two-dimensional problem of continuum percolation on the surface of a Klein
bottle have been calculated by the Monte Carlo method with the Newman--Ziff
algorithm for completely permeable disks. It has been shown that the
percolation threshold of disks on the Klein bottle coincides with the
percolation threshold of disks on the surface of a torus, indicating that this
threshold is topologically invariant. The scaling exponents determining
corrections to the wrapping probability and critical concentration owing to the
finite-size effects are also topologically invariant. At the same time, the
quantities $R_{\infty}$ are different for percolation on the torus and Klein
bottle and are apparently determined by the topology of the surface.
Furthermore, the difference between the $R_{\infty}$ values for the torus and
Klein bottle means that at least one of the percolation clusters is degenerate.
