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Explore Stack InternalA very nice example for (4) is a "long circle". That is, let $L$ be the closed long line, with endpoints $0$ and $\omega_1$, and adjoin to $L$ a path from $0$ to $\omega_1$ to get a space $S$ (or equivalently, you could just identify $0$ and $\omega_1$ together). Then $S$ is path-connected and locally connected (and also compact and Hausdorff), but it is not locally path-connected at $\omega_1$.
A very nice example for (4) is a "long circle". That is, let $L$ be the closed long line, with endpoints $0$ and $\omega_1$, and adjoin to $L$ a path from $0$ to $\omega_1$ to get a space $S$. Then $S$ is path-connected and locally connected, but it is not locally path-connected at $\omega_1$.
A very nice example for (4) is a "long circle". That is, let $L$ be the closed long line, with endpoints $0$ and $\omega_1$, and adjoin to $L$ a path from $0$ to $\omega_1$ to get a space $S$ (or equivalently, you could just identify $0$ and $\omega_1$ together). Then $S$ is path-connected and locally connected (and also compact and Hausdorff), but it is not locally path-connected at $\omega_1$.
A very nice example for (4) is a "long circle". That is, let $L$ be the closed long line, with endpoints $0$ and $\omega_1$, and adjoin to $L$ a path from $0$ to $\omega_1$ to get a space $S$. Then $S$ is path-connected and locally connected, but it is not locally path-connected at $\omega_1$.
