Definition:Weakly Locally Connected at Point/Definition 2

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

The space $T$ is weakly locally connected at $x$ every open neighborhood $U$ of $x$ contains an open neighborhood $V$ of $x$ such that every two points of $V$ lie in some connected subset of $U$.

Also see

 * Equivalence of Definitions of Weakly Locally Connected at Point