A matrix is totally positive, respectively nonnegative, if all its square minor determinants are positive, respectively nonnegative, and the matrix is Toeplitz if its entries are constant along diagonals parallel to the main diagonal. In this talk we show that every totally nonnegative Toeplitz matrix of a given size, say $m$ $\times$ $n$, is the limit of totally positive mxn Toeplitz matrices, or in other words, that the closure of the set of totally positive $m$ $\times$ $n$ Toeplitz matrices, in the euclidean space of all $m$ $\times$ $n$ matrices, is equal the set of totally nonnegative Toeplitz matrices. The proof follows from a characterization of totally positive/ nonnegative Toeplitz matrices in terms of certain mixed Hodge-Riemann relations obtained in joint work with P. Macias Marques, A. Seceleanu, and J. Watanabe. Analogous results have been previously obtained, for arbitrary matrices by A. Whitney (1952), and more recently, for Hankel matrices by Fallet-Johnson-Sokal (2017, 2021).