Remark 2.1 For the application of the results, it is possible to take the initial moment as t$_{0}$ = 0 , without being the discontinuity moment since of the group property. Then x$_{0}$ / ∈ Γ ∪ ˜ Γ . Denote by ̂ [ a, b ] , a, b ∈ $^{R}$, the interval [ a, b ] , whenever a ≤ b and [ b, a ] , otherwise. Let x$_{1}$ ( t ) ∈ P C ( $^{R$_{+}$}$, θ $^{1}$) , θ 1 = { θ 1 i } , and x$_{2}$ ( t ) ∈ P C ( $^{R$_{+}$}$, θ $^{2}$) , θ 2 = { θ 2 i } , be two different solutions of (2.1). Definition 2.3 The solution x$_{2}$ ( t ) is in the ϵ − neighborhood of x$_{1}$ ( t ) on the interval I if The topology defined with the help of ϵ − neighborhoods is called the B-topology. It can be apparently seen that it is Hausdorff and it can be considered also if two solutions x$_{1}$ ( t ) and x$_{2}$ ( t ) are defined on a semi-axis or on the entire real axis. Definition 2.4 The solution x$_{0}$ ( t ) = x ( t, 0 , x$_{0}$ ) , t ∈ $^{R}$, x$_{0}$ ∈ D, of (2.1) B-continuously depends on x$_{0}$ for increasing t if there corresponds a positive number δ to any positive ϵ and a finite interval [0 , b ] , b > 0 such that any other solution x ( t ) = x ( t, 0 , ˜ x ) of (2.1) lies in ϵ − neighborhood of x$_{0}$ ( t ) on [0 , b ] whenever ˜ x ∈ B ( x$_{0}$, δ ) . Similarly, the solution x$_{0}$ ( t ) of (2.1) B-continuously depends on x$_{0}$ for decreasing t if there corresponds a positive number δ to any positive ϵ and a finite interval [ a, 0] , a < 0 such that any other solution x ( t ) = x ( t, 0 , ˜ x ) of (2.1) lies in ϵ − neighborhood of x$_{0}$ ( t ) on [ a, 0] whenever ˜ x ∈ B ( x$_{0}$, δ ) . The solution x$_{0}$ ( t ) of (2.1) B-continuously depends on x$_{0}$ if it continuously depends on the initial value, x$_{0}$, for both increasing and decreasing t. If conditions (C1)-(C7) hold, then each solution x$_{0}$ ( t ) : $^{R}$→ R $^{n}$, x$_{0}$ ( t ) = x ( t, 0 , x$_{0}$ ) , of (2.1) continu- ously depends on x$_{0}$ [1]. In order to facilitate the analysis of the system with variable moments of impulses (2.1), a B-equivalent system [1] to the system with variable moments of impulses will be utilized in our study. Below, we will construct the B-equivalent system. Let x ( t ) = x ( t, 0 , x$_{0}$ + Δ x ) be a solution of system (2.1) neighbor to x$_{0}$ ( t ) with small ‖ Δ x ‖ . If the point x$_{0}$ ( θ$_{i}$ ) is a ( β ) − or ( γ ) − type point, then it is a boundary point. For this reason, there exist two different possibilities for the near solution x ( t ) with respect to the surface of discontinuity. They are: • the sets θ 1 and θ 2 have same number of elements in I ; • | θ 1 i − θ 2 i | < ϵ for all θ 1 i ∈ I ; • the inequality || x$_{1}$ ( t ) − x$_{2}$ ( t ) || < ϵ is valid for all t, which satisfy t ∈ I \ ∪ θ 1 i ∈ I ( θ 1 i − ϵ, θ 1 i + ϵ ) . ( N 1) The solution x ( t ) intersects the surface of discontinuity, Γ , at a moment near to θ$_{i}$, ( N 2) The solution x ( t ) does not intersect Γ , in a small time interval centered at θ$_{i}$. 2.1 B-equivalence to a system with fixed moments of impulses 6