Equivalence of Definitions of Locally Connected Space/Definition 4 implies Definition 3

$(4)$ implies $(3)$
Let $T = \struct{S, \tau}$ be locally connected by Definition 4:
 * the components of the open sets of $T$ are also open in $T$.

Let $\mathcal B = \{ U \in \tau : U$ is connected in $T \}$.

Let $U$ be open in $T$.

By assumption, the components of $U$ are open in $T$.

From Connected Set in Subspace, the components of $U$ are connected in $T$.

By the definition of the components of a topological space, $U$ is the union of its components.

Hence $U$ is the union of open connected sets in $T$.

By definition, $\mathcal B$ is an basis for $\tau$.

That is, $T = \struct{S, \tau}$ is locally connected by Definition 3.

Also see

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