Published online by Cambridge University Press: 09 April 2009
In a recent paper [4], H. Sharp, Jr., has discussed the problem of finding formulae for the following naturally defined integers: the numbers t(n), tc(n), t0(n), tc0(n), and ts(n) of all homeomorphism classes of finite topological spaces with n elements, which are respectively (i) arbitrary, (ii) connected, (iii) T0, (iv) connected and T0, (v) symmetric. Here, a finite topological space X is called symmetric provided the following relation ≦ is symmetric: x ≦ y if and only if x ∈ Uv, the intersection of all open sets containing y.