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For the classical subgroup-covering construction, assume $X$ is path-connected, locally path-connected, and semilocally simply connected. These are the usual hypotheses underlying the classification of connected covering spaces; an arbitrary space, as literally allowed by the wording, need not admit every subgroup covering.

Construct the path-class construction of a subgroup covering directly. Consider paths $\alpha$ starting at $x_0$, and identify $\alpha ,\beta$ when they have the same endpoint and the loop class $[\alpha {\beta }^{-1}]$ belongs to $H$. The subgroup properties make this an equivalence relation. Let $Y$ be its set of classes and define $p([\alpha ])=\alpha (1)$.

Choose a path-connected open neighbourhood $U$ of $\alpha (1)$ whose loops become null-homotopic in $X$. Such neighbourhoods form a local basis under the stated local hypotheses. Define a sheetFor each point of $U$ the resulting class is independent of the chosen path $\eta$, since the difference is a loop in $U$, conjugated by $\alpha$, and is trivial in ${\pi }_1(X,x_0)$. Thus $p$ maps this sheet bijectively onto $U$. Give $Y$ the topology generated by these sheets. Restrictions to smaller such neighbourhoods show that the sheets form a basis and that their endpoint maps are homeomorphisms. Two sheets over $U$ that intersect coincide, by concatenating and reversing paths within $U$; otherwise they are disjoint. Therefore $p^{-1}(U)$ is a disjoint union of copies of $U$, and $p$ is a covering map.

The basepoint is the class of the constant path. The path $\alpha$ lifts to its successive prefix classes, giving a path from this basepoint to $[\alpha ]$; hence $Y$ is path-connected. A loop $\gamma$ lifts closed exactly when its final prefix class agrees with the constant class, namely when $[\gamma ]\in H$. Consequently $p_{\ast }({\pi }_1(Y))=H$. Part (i) supplies injectivity, so

The missing local hypothesis is substantive. In the Hawaiian earring, every neighbourhood of the common basepoint contains a whole sufficiently small circle. Its loop is nontrivial in the whole space, since retracting onto that circle preserves it. An evenly covered neighbourhood would lift that loop to a closed loop in one sheet, so the image subgroup of any covering contains a nontrivial element. Therefore the trivial subgroup cannot be realized by a simply connected covering of this space. The intended result is valid under the local covering-space hypotheses above, not for every topological space.