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

of subsets of a set X we introduce different convexities and topologies on both

\mathcal{H} H

and X. Then we prove that the convexity

\mathcal{G} G

generated by

\mathcal{H} H

is of arity N whenever

\mathcal{H} H

is connected and X is N-connected.