The quasitopological fundamental group π qtop 1 (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 π qtop 1 (X, x 0 ) and properties of the underlying space X such as 'π 1 -shape injectivity' and 'homotopically path-Hausdorff. ' A space X is π 1 -shape injective if the fundamental group canonically embeds in the first shape group so that the elements of π 1 (X, x 0 ) can be represented as sequences in an inverse limit. We show a locally path connected metric space X is π 1 -shape injective if and only if π qtop 1 (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 π 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 π qtop 1 (X, x 0 ) satisfies the T 1 separation axiom. qtop 1 (X, x 0 ). Examples [6] illustrate that π qtop 1 (X, x 0 ) need not be a topological group even if X is a compact metric space [20][21]. 1 arXiv:1304.6453v3 [math.AT] 26 Jun 2013 π qtop 13. Corollary 33: If X is both locally path connected and the inverse limit of nested retracts of polyhedra, then π qtop 1 (X, x 0 ) is T 1 k s + 3 π qtop 1 (X, x 0 ) is T 2 k s + 3 X is π 1 -shape injective. Theorem 34: If πqtop 1 (X, x 0 ) is T 3 , then π qtop 1 (X, x 0 ) is T 4 .

The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a new example of a quotient map such that q × q fails to be a quotient map. This provides a counterexample to the question of whether the fundamental group (with the quotient topology) of a compact metric space is always a topological group with the standard operations.

We develop a new route through which to explore ker Ψ X , the kernel of the π 1 −shape group homomorphism determined by a general space X, and establish, for each locally path connected, paracompact Hausdorff space X, ker Ψ X is precisely the Spanier group of X.

The fundamental group of a locally path connected metric space inherits the discrete topology in a natural way if and only if X is semilocally simply connected. We also provide a counterexample to a similar theorem in the literature.

Endowed with natural topologies, the fundamental group of the Hawaiian earring continuously injects into the inverse limit of free groups. This note shows the injection fails to have a continuous inverse. Such a phenomenon was unexpected and appears to contradict results of another author.