This article derives a set of ”easily verifiable” sufficient conditions for the local asymptotic stability of the trivial solution of \[ (1.1)\quad \frac{dx(t)}{dt}+ax(t)+[x^*+x(t)]\sum^{\infty}_{j=1}b_ jx(t-\tau_ j)=0\quad (t>0) \] and then examines the ”size” of the domain of attraction of the trivial solution of the nonlinear system (1.1) with a countable number of discrete delays. The following assumptions are made: parameters \(x^*,b_ j,\tau_ j\) \((j=1,2,...)\) are positive constants such that \(\sum^{\infty}_{j=1}b_ j=b<\infty\); \(0<\inf \tau_ j=\tau_*\leq \sup_{j} \tau_ j=\tau^*<\infty\) and a is a nonnegative constant. The following theorem gives a sufficient condition for the local asymptotic stability. Theorem 2.1. Assume that: (i) a is a nonnegative constant in (1.1); (ii) \(x^*,b_ j,\tau_ j\) \((j=1,2,...)\) are positive constants such that \(\sum^{\infty}_{j=1}b_ j<\infty\); \(0<\inf_{j} \tau_ j=\tau_*\leq \sup_{j} \tau_ j=\tau^*<\infty\); (iii) \(r^*x^*\{\sum^{\infty}_{j=1}b_ j\}<\pi /2\). Then the trivial solution of (1.1) is locally asymptotically stable. Define the entire set S of initial conditions of (1.1) as (3.1) \(S=\{\phi | \phi (s)\geq - x^*\); \(s\in [-\tau^*,0]\); \(\phi (0)>-x^*\); \(\phi\) is continuous on \([-\tau^*,0]\}\). The main result is the following theorem. Theorem 3.1. Assume that the conditions of Theorem 2.1 hold. Then the domain of attraction of the trivial solution of (1.1) is the entire set S defined in (3.1).