The Hawaiian earring is locally path-connected but has no universal cover

Given: The objects, hypotheses, and choice principles stated above.

The loop $t\mapsto [t]$ in $\mathbb{R}\mathrm{/}\mathbb{Z}$ is not nullhomotopic. (The projected unit interval is not nullhomotopic in $\mathbb{R}\mathrm{/}\mathbb{Z}$).

If a space admits a universal covering, then it is semilocally simply connected. No local path-connectedness hypothesis is required. (A space admitting a universal covering is semilocally simply connected).

A space $X$ is semilocally simply connected at $x\in X$ when there is a neighbourhood $U$ of $x$ and a basepoint-preserving inclusion $(U,x)\hookrightarrow (X,x)$ whose induced map on fundamental groups is trivial (def-neighbourhood-top, def-induced-homomorphism-on-fundamental-groups, def-based-loops-and-fundamental-group). It is semilocally simply connected when this holds at every point. The neighbourhood need not itself be simply connected. (Semilocally simply connected spaces with explicit basepoint convention).

The quotient topology. Let $(X,\mathcal{T})$ be a topological space (def-topological-space), let $Y$ be a set and let $q:X\to Y$ be a surjection (def-injection-surjection-bijection). The quotient topology on $Y$ induced by $q$ is the final topology of the one-element family $(q)$ (def-initial-and-final-topology): ${\mathcal{T}}_q:=\{V\subseteq Y:q^{-1}[V]\in \mathcal{T}\}\mathrm{.}$ That this is a topology is discharged in def-initial-and-final-topology, where every final topology is verified to satisfy (T1), (T2) and (T3). Dually, $C\subseteq Y$ is closed in ${\mathcal{T}}_q$ exactly when $q^{-1}[C]$ is closed in $X$, because $q^{-1}[Y\setminus V]=X\setminus q^{-1}[V]$. (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).