The classification of the reducing subspaces of analytic Toeplitz operators on the classical Hardy space $H^2$ was completed in the 1970s due to work by Cowen and by Thomson. As for the reducing subspaces of non-analytic Toeplitz operators, to the best of our knowledge, there is no result in the literature so far.
We initiate to describe the reducing subspaces of Toeplitz operators via an approximation approach, showing that in the class of Toeplitz operators with continuous symbols those irreducible ones constitute a dense $G_\delta$. Our result depends on a finite-dimensional approximation result, which asserts that in the class of $n\times n$ Toeplitz matrices those irreducible ones constitute an open dense subset.