Definition:Locally Connected Space/Definition 2

Definition
A topological space $T = \struct{S, \tau}$ is locally connected $T$ is weakly locally connected at each point of $T$.

That is, a topological space $T = \struct{S, \tau}$ is locally connected each point of $T$ has a neighborhood basis consisting of connected sets of $T$.

Also see

 * Equivalence of Definitions of Locally Connected Space