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

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Theorem

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

Let $T$ have a basis $\mathcal B$ 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:

$\mathcal B_x = \set{B \in \mathcal B: x \in B}$

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 path-connected sets.

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

$\blacksquare$