A core-free semicovering of the Hawaiian Earring

Abstract

The connected covering spaces of a connected and locally path-connected topological space $X$ can be classified by the conjugacy classes of those subgroups of $\pi_1(X,x)$ which contain an open normal subgroup of $\pi_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring $\mathds{H}$ with corresponding open subgroup of $\pi_1(\mathds{H})$ which does not contain {\em any} nontrivial normal subgroup of $\pi_1(\mathds{H})$.