Let A A be a TAF-algebra, Z ( A ) Z(A) the centre of A , I d ( A ) A, Id(A) the ideal lattice of A A , and M i r ( A ) Mir(A) the space of meet-irreducible elements of I d ( A ) Id(A) , equipped with the hull-kernel topology. It is shown that M i r ( A ) Mir(A) is a compact, locally compact, second countable, T 0 T_0 -space, that I d ( A ) Id(A) is an algebraic lattice isomorphic to the lattice of open subsets of M i r ( A ) Mir(A) , and that Z ( A ) Z(A) is isomorphic to the algebra of continuous, complex functions on M i r ( A ) Mir(A) . If A A is semisimple, then Z ( A ) Z(A) is isomorphic to the algebra of continuous, complex functions on P r i m ( A ) Prim(A) , the primitive ideal space of A A . If A A is strongly maximal, then the sum of two closed ideals of A A is closed.