  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.
