In this talk we show that for a continuum X, the following conditions are equivalent: (i) the continuum X is locally connected, (ii) each non-cut set of X has arbitrarily small open neighborhoods whose complements are connected, (iii) each non-cut set of X has continuum-wise connected complement, (iv) the continuum X is aposyndetic with respect to each of its non-cut sets, and (v) the continuum X is aposyndetic with respect to each of its nonempty closed sets.

Co-authors: Raúl Escobedo and Eduardo García-Muñoz.