Locally Path-Connected Space is not necessarily Path-Connected
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Then it is not necessarily the case that $T$ is also path-connected.
Let $a, b, c \in \R$ where $a < b < c$.
- $A := \openint a b \cup \openint b c$
From Union of Adjacent Open Intervals is not Path-Connected, $T$ is not a path-connected space.
Hence the result.