Summary: Consider the equation \(G(x)=\tilde{y}\), where the mapping \(G\) acts from a metric space \(X\) into a space \(Y\), on which a distance is defined, \( \tilde{y} \in Y\). The metric in \(X\) and the distance in \(Y\) can take on the value \(\infty\), the distance satisfies only one property of a metric: the distance between \(y, z \in Y\) is zero if and only if \(y=z\). For mappings \(X \to Y\) the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space \(X\) of solutions of the considered equation to changes of the mapping \(G\) and the element \(\tilde{y}\). This assertion is applied to the study of the integral equation \[f \left(t, \int_0^1 \mathcal{K}(t,s)x(s) ds, x(t)\right)=\tilde{y}(t), \quad t \in [0.1],\] with respect to an unknown Lebesgue measurable function \(x: [0,1] \to \mathbb{R}\). Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions \(f, \mathcal{K}, \tilde{y}\).