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

$(3)$ implies $(4)$
Let $T = \struct{S, \tau}$ be locally connected by Definition 3:
 * $T$ has a basis consisting of connected sets in $T$.

Let $U$ be an open subset of $T$.

From Open Set is Union of Elements of Basis, $U$ is a union of open connected sets in $T$.

From Open Set in Open Subspace and Connected Set in Subspace, $U$ is a union of open connected sets in $U$.

From Components are Open iff Union of Open Connected Sets, the components of $U$ are open in $U$.

From Open Set in Open Subspace then the components of $U$ are open in $T$.

That is, $T = \struct{S, \tau}$ is locally connected by Definition 4.

Also see

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