Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 1

$(3)$ implies $(1)$
Let $T = \struct{S, \tau}$ be locally connected by Definition 3:
 * $T$ has a basis consisting of connected sets in $T$.

For each $x \in S$ we define:
 * $\mathcal B_x = \left\{{B \in \mathcal B: x \in B}\right\}$

From Basis induces Local Basis, $\mathcal B_x$ is a local basis.

As each element of $\mathcal B_x$ is also an element of $\mathcal B$, it follows that $\mathcal B_x$ is also formed of connected sets.

Thus, for each point $x \in S$, there is a local basis which consists entirely of connected sets.

That is, $T = \struct{S, \tau}$ is locally connected by Definition 1.

Also see

 * Equivalence of Definitions of Locally Path-Connected Space, whose proof is almost the same