Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1

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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 $\mathcal B_x = \set{ W \in \tau : x \in W, W \text{ is path-connected in } T}$.

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 $V \in \mathcal B_x$.

Because $U$ was arbitrary, $\mathcal B_x$ 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 $T$.

$\blacksquare$