Recently, we have introduced the concept of unpredictable points and motions for dynamical systems [4]. Next, an unpredictable function was defined as an unpredictable point of the Bebutov dynamics, and first theorems on the existence of unpredictable solutions were proved in [5, 6]. The metric of the Bebutov dynamics is not convenient for applications, since it is hard to verify. For this reason, in the present study, we apply the topology of convergence on compact sets to define unpredictable functions. The topology is metrizable and easy for applications with integral operators. Thus, one can accept that we lay a corner stone to the foundation of differential equations theory related to unpredictable solutions, and consequently, chaos. 5 Conclusion References [1] Akhmet, M. U., Dynamical synthesis of quasi-minimal sets, Int. J. Bifurcat. Chaos 19 , 2423–2427, 2009. [2] Akhmet, M. U. & Fen, M. O., Replication of chaos, Commun. Nonlinear Sci. Numer. Simulat. 18 , 2626–2666, 2013. [3] Akhmet, M. & Fen, M. O., Replication of Chaos in Neural Networks, Economics and Physics , Springer- Verlag, Berlin, Heidelberg, 2016. [4] Akhmet, M. & Fen, M. O., Unpredictable points and chaos, Commun. Nonlinear Sci. Numer. Simulat. 40 , 1–5, 2016. [5] Akhmet, M. & Fen, M. O., Existence of unpredictable solutions and chaos, Turk. J. Math. , in press, doi:10.3906/mat-1603-51. [6] Akhmet, M. & Fen, M. O., Poincaré chaos and unpredictable functions, arXiv:1608.07301, submitted. [7] Devaney, R. L., An Introduction to Chaotic Dynamical Systems , Addison-Wesley Publishing Company, United States of America, 1987. [8] Driver, R. D., Ordinary and Delay Differential Equations , Springer, New York, 1977. [9] Hale, J. & Koçak, H., Dynamics and Bifurcations , Springer-Verlag, New York, 1991. [10] Hammel, S. M., Yorke, J. A. & Grebogi, C., Do numerical orbits of chaotic dynamical processes represent true orbits?, J. Complexity 3 , 136–145, 1987. [11] Horn, R. A. & Johnson, C. R., Matrix Analysis , Cambridge University Press, United States of America, 1992. [12] Li, T. Y. & Yorke, J. A., Period three implies chaos, Am. Math. Mon. 82 , 985–992, 1975. 11