Hawaiian earring

Example

For $n \ge 1$ let $C_n \subseteq \R^2$ be the circle with centre $(1/n, 0)$ and radius $1/n$. The Hawaiian earring is $H = \bigcup_{n \ge 1} C_n$ with the subspace topology from $\R^2$. All circles pass through the origin $0$. The space $H$ is compact and path-connected, but it is not semilocally simply connected at $0$.

Check. $H$ is closed and bounded in $\R^2$, so it is compact. Each $C_n$ is path-connected and contains $0$, so $H$ is path-connected. Fix $n$ and let $r_n \colon H \to C_n$ be the identity on $C_n$ and send every other circle to $0$. This map is continuous, and it is a retraction onto $C_n$. Let $\gamma_n$ be the loop that goes once around $C_n$. The map $r_n$ sends it to a generator of the fundamental group $\pi_1(C_n) \cong \Z$, so $\gamma_n$ is not null-homotopic in $H$. Every neighbourhood of $0$ contains $C_n$ for all large $n$, so it contains loops at $0$ that are not null-homotopic in $H$.