In this paper we consider Aronszajn’s-type topological charac- terization (or compact $R_{\delta}$ property) of the set of solutions to the following integral equation $$x(t)=V\left(t, x(\theta_1(t)), \int_0^t F\left(t, s, x(\theta_2(s)), \int_0^t r(s, \tau)x(\theta_3(\tau))d\tau\right)\right)+\int_0^t K(t, s)g(s, x(\theta_4(s)))ds,$$ where $t\in [0,\infty);$ $\theta_i: [0,\infty)\to [0,\infty),$ $i=1, 2, 3, 4;$ $K: [0,\infty) \times [0,\infty) \to L(E,E);$ $V: [0,\infty) \times E \times E\to E$; $F: [0,\infty) \times [0,\infty) \times E \times E\to E$; $r: [0,\infty)\times [0,\infty) \to R$; $g: [0,\infty)\times E\to E$; $E$ is a real Banach space with norm $|.|$; $L(E,E)$ the Banach space of continuous linear operators with domain $E$ and range in $E$.