Let C[a,b] be the space of all real-valued continuous functions on the finite closed interval [a,b] and M a given n-dimensional linear subspace of C[a,b]. For this subspace M, we put the following two subsets of C[a,b]: \(S_ M=\{f| f\) possesses a unique best minimax approximation from \(M\}\), \(A_ M=\{g|\) the error function \(e=g-\tilde g\) has an alternating set of \((n+1)\) points in [a,b] for any best minimax approximation \(\tilde g\) to g from \(M\}\). In this paper, by using an inclusion relation between \(S_ M\) and \(A_ M\), we give a characterization for the space M to be a Chebyshev space.