It seems indeed pretty clearly to have been Riemann, in §6 of his Inaugural Dissertation (1851). There he defines zusammenhängend as well as einfach, zweifach and mehrfach zusammenhängend without citing any prior sources.

Yet a possible antecedent is Gauss, of whom J.-C. Pont's book (cited in the comments above and reviewed here) singles out a text, written around 1840, "which one can consider as a sketch of the theory of the order of connectivity".

(On Gauss and Riemann see also a letter of Betti published by A. Weil, Riemann, Betti, and the Birth of Topology, Archive for History of Exact Science 20 (1979) 91-96.)

It seems indeed pretty clearly to have been Riemann, in §6 of his Inaugural Dissertation (1851). There he defines zusammenhängend as well as einfach, zweifach and mehrfach zusammenhängend without citing any prior sources.

Yet a possible antecedent is Gauss, of whom J.-C. Pont's book (cited in the comments above and reviewed here) singles out a text, written around 1840, "which one can consider as a sketch of the theory of the order of connectivity". It would be interesting to dig Pont's exact reference, which ought to be available in Gauss's Werke.

(On Gauss and Riemann see also a letter of Betti published by A. Weil, Riemann, Betti, and the Birth of Topology, Archive for History of Exact Science 20 (1979) 91-96.)