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

Theorem

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

Let $T$ have a basis consisting of connected sets in $T$.

Then the components of the open sets of $T$ are also open in $T$.

Proof

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 $T$.

$\blacksquare$