Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1
Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let each point of $T$ have a neighborhood basis 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
Let $x \in S$ and $x \in U \in \tau$.
Let $\BB_x = \set {W \in \tau : x \in W, W \text { is path-connected in } T}$.
By definition of local basis, we have to show that there exists a path-connected open set $V \in \tau$ with $x \in V \subset U$.
Let $V = \map {\operatorname {PC}_x} U$ denote the path component of $x$ in $U$.
From Open Set in Open Subspace, it suffices to show that $V$ is open in $U$.
By Set is Open iff Neighborhood of all its Points, we may do this by showing that $V$ is a neighborhood in $U$ of all of its points.
Let $y \in V$.
By assumption, there exists a path-connected neighborhood $W$ of $y$ in $T$ with $W \subset U$.
By Neighborhood in Topological Subspace, $W$ is a neighborhood of $y$ in $U$.
By definition of path component:
- $W \subset \map {\operatorname {PC}_y} U = \map {\operatorname {PC}_x} U = V$
By Neighborhood iff Contains Neighborhood, $V$ is a neighborhood of $y$ in $U$.
Because $y$ was arbitrary, $V$ is open in $U$.
From Open Set in Open Subspace, $V$ is open in $T$.
Hence $V \in \BB_x$.
Because $U$ was arbitrary, $\BB_x$ is a local basis of $x$ consisting of (open) path-connected sets.
Since $x$ was arbitrary, then each point of $T$ has a local basis consisting entirely of path-connected sets in $T$.
$\blacksquare$