Let X be an infinite regular topological space which is connected and locally connected. If no point of X is a cut point, does X always have base of connected open sets whose complements (with respect to X) are also connected? The answer is "YES" if X is a real Banach space whose dimension is at least 2. The motivation for this question is to obtain a simple characterization of as large a class of spaces as possible, which have the property that their topology is uniquely determined when the collection of their connected subsets is specified. Note that closed connected sets can be defined in terms of connected sets. They are those connected sets whose union with the singleton of any point not belonging to them is not connected. Then we can take the complements of the closed connected sets to be the base of our topology.