“…In Section 2, we set some preliminary notation and introduce the topological spaces which will be used throughout the rest of the work. Particularly, we recall the definitions of the spaces of Lipschitz Carathéodory (LC) and Strong Carathéodory (SC) functions, and endow them with several possible topologies: besides the already established topologies (which are collectively analyzed in [15,16]), such as the classic T B , T D (firstly presented in [17,18]) and σ D (also treated in [2,3,4,12,19]), and the more recent T Θ and σ Θ (firstly presented in [15] and [16], respectively), we introduce the new hybrid topologies T ΘD , σ ΘD , T Θ Θ , σ Θ Θ and T ΘB , where by hybrid we mean that they are, somehow, derived from the previous ones in a way that it is possible to treat the first N components of the spatial variable (representing the current state in a delay differential equation) in a different way from the last N ones (representing the history of the state). The symbols Θ and Θ stand for sets of moduli of continuity which identify numerable quantities of compact sets of continuous functions on which the sequences of elements in LC are asked to converge.…”