Subbases for N-ary Convexities

We introduce a notion of N-connectedness of a topological space with respect to a convexity on this space. For a given collection $\mathcal{H}$ of subsets of a set X we introduce different convexities and topologies on both $\mathcal{H}$ and X. Then we prove that the convexity $\mathcal{G}$ generated by $\mathcal{H}$ is of arity N whenever $\mathcal{H}$ is connected and X is N-connected.