On fundamental groups with the quotient topology

The quasitopological fundamental group ${\pi }_1^{qtop}(X,x_0)$ is the fundamental group endowed with the natural quotient topology inherited from the space of based loops and is typically non-discrete when $X$ does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group ${\pi }_1^{qtop}(X,x_0)$ and properties of the underlying space $X$ such as $\pi_{1}$-shape injectivity' and homotopically path-Hausdorff.’ A space $X$ is ${\pi }_1$-shape injective if the fundamental group canonically embeds in the first shape group so that the elements of ${\pi }_1(X,x_0)$ can be represented as sequences in an inverse limit. We show a locally path connected metric space $X$ is ${\pi }_1$-shape injective if and only if ${\pi }_1^{qtop}(X,x_0)$ is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that $X$ is not ${\pi }_1$-shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space $X$ is homotopically path-Hausdorff if and only if ${\pi }_1^{qtop}(X,x_0)$ satisfies the $T_1$ separation axiom.