Lacunar ring topologies and maximum ring topologies with a prescribed convergent sequence

We show a ring topology on k[X], which we call lacunar, whose completion is a subring of k[[X]] consisting of some particular series with many zero coefficients. Zelenyuk, Protasov and Khromulyak have proven that each countable ring is complete with the ring topology which is maximum among those for which a certain T-sequence converges. We present some concrete examples of this situation on the rings k[X], image, k(X) and image.