Locally Path-Connected Space is not necessarily Path-Connected

Theorem
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.

Then it is not necessarily the case that $T$ is also path-connected.

Proof
Let $T := \struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.

Let $a, b, c \in \R$ where $a < b < c$.

Let $A$ be the union of the two open intervals:
 * $A := \openint a b \cup \openint b c$

Let $T' := \struct {A, \tau_A}$ be the subspace composed of $A$ with the subspace topology induced on $\tau_d$ by $A$.

From Union of Adjacent Open Intervals is Locally Path-Connected, $T'$ is a locally path-connected space.

From Union of Adjacent Open Intervals is not Path-Connected, $T$ is not a path-connected space.

Hence the result.