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

Theorem

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

Let $T$ have a basis $\BB$ consisting of path-connected sets in $T$.

Then

each point of $T$ has a local basis consisting entirely of path-connected sets in $T$.

Proof

For each $x \in S$ we define:

$\BB_x = \set {B \in \BB: x \in B}$

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

As each element of $\BB_x$ is also an element of $\BB$, it follows that $\BB_x$ is also formed of path-connected sets.

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

$\blacksquare$